For the subspace $C_k(\omega_1;{\mathbb Z})$ of $C_k(\omega_1)$ consisting of integer-valued functions, the proof of the Lindel"of property is relatively simple.

Given any open cover ${\mathcal U}$ of $C_k(\omega_1;{\mathbb Z})$, for every $f\in C_k(\omega_1;{\mathbb Z})$ find a countable ordinal $\alpha_f$ such that $f|[\alpha_f,\omega_1)$ is constant and the set $B[f;\alpha_f]=\{g\in C_k(\omega_1;{\mathbb Z}):g|[0,\alpha_f]=f|[0,\alpha_f]\}$ is contained in some neighborhood $U_f\in{\mathcal U}$ of $f$. For every ordinal $\alpha\in\omega_1$ consider the subspace $C_k(\alpha;{\mathbb Z})$ of $C_\omega(\omega_1;{\mathbb Z})$ consisting of functions that are constant on the interval $[\alpha,\omega_1)$. It is easy to see that the subspace $C_k(\alpha;{\mathbb Z})$ is Lindel"of (moreover, countable). Put $\alpha_0=0$ and construct an increasing sequence $(\alpha_n)_{n\in\omega}$ of countable ordinals as follows. Assume that a countable ordinal $\alpha_n$ has been constructed. Consider the open cover $\{B[f;\alpha_f]:f\in C_k(\alpha_n;{\mathbb Z})\}$ of the Lindel"of subspace $C_k(\alpha_n;{\mathbb Z})$ and find a countable subset $F_n\subset C_k(\alpha_n;{\mathbb Z})$ such that $C_k(\alpha_n,{\mathbb Z})\subset\bigcup_{f\in F_n}B[f_n;\alpha_{f_n}]$. Let $\alpha_{n+1}=\sup\{\alpha_f+1:f\in F_n\}$.

We claim that for the countable set $F=\bigcup_{n\in\omega}F_n$ the family ${\mathcal U}_F:=\{U_f:f\in F\}\subset {\mathcal U}$ is a required countable subcover of $C_k(\omega_1;{\mathbb Z})$. Given any function $f\in C_k(\omega_1;{\mathbb Z})$, use the continuity of $f$ at the ordinal $\alpha_\omega=\sup_{n\in\omega}\alpha_n$ and find $n\in\omega$ such that $f([\alpha_n,\alpha_\omega])=\{f(\alpha_\omega)\}$. Find a unique function $g\in C(\alpha_n;{\mathbb Z})$ such that $g|[0,\alpha_n]=f|[0,\alpha_n]$ and observe that $g|[0,\alpha_\omega]=f|[0,\alpha_\omega]$. Since $\alpha_g<\alpha_{n+1}$, we get $f\in B[g;\alpha_g]\subset U_g\in{\mathcal U}_F$.

Now it is necessary to adapt this proof to the general case of the space $C_k(\omega_1)$ of all (not necessarily integer-valued) functions on $\omega_1$.

For the subspace $C_k(\omega_1;{\mathbb Z})$ of $C_k(\omega_1)$ consisting of integer-valued functions, the proof of the Lindelöf property is relatively simple.

Given any open cover ${\mathcal U}$ of $C_k(\omega_1;{\mathbb Z})$, for every $f\in C_k(\omega_1;{\mathbb Z})$ find a countable ordinal $\alpha_f$ such that $f|[\alpha_f,\omega_1)$ is constant and the set $B[f;\alpha_f]=\{g\in C_k(\omega_1;{\mathbb Z}):g|[0,\alpha_f]=f|[0,\alpha_f]\}$ is contained in some neighborhood $U_f\in{\mathcal U}$ of $f$. For every ordinal $\alpha\in\omega_1$ consider the subspace $C_k(\alpha;{\mathbb Z})$ of $C_\omega(\omega_1;{\mathbb Z})$ consisting of functions that are constant on the interval $[\alpha,\omega_1)$. It is easy to see that the subspace $C_k(\alpha;{\mathbb Z})$ is Lindelöf (moreover, countable). Put $\alpha_0=0$ and construct an increasing sequence $(\alpha_n)_{n\in\omega}$ of countable ordinals as follows. Assume that a countable ordinal $\alpha_n$ has been constructed. Consider the open cover $\{B[f;\alpha_f]:f\in C_k(\alpha_n;{\mathbb Z})\}$ of the Lindelöf subspace $C_k(\alpha_n;{\mathbb Z})$ and find a countable subset $F_n\subset C_k(\alpha_n;{\mathbb Z})$ such that $C_k(\alpha_n,{\mathbb Z})\subset\bigcup_{f\in F_n}B[f_n;\alpha_{f_n}]$. Let $\alpha_{n+1}=\sup\{\alpha_f+1:f\in F_n\}$.

We claim that for the countable set $F=\bigcup_{n\in\omega}F_n$ the family ${\mathcal U}_F:=\{U_f:f\in F\}\subset {\mathcal U}$ is a required countable subcover of $C_k(\omega_1;{\mathbb Z})$. Given any function $f\in C_k(\omega_1;{\mathbb Z})$, use the continuity of $f$ at the ordinal $\alpha_\omega=\sup_{n\in\omega}\alpha_n$ and find $n\in\omega$ such that $f([\alpha_n,\alpha_\omega])=\{f(\alpha_\omega)\}$. Find a unique function $g\in C(\alpha_n;{\mathbb Z})$ such that $g|[0,\alpha_n]=f|[0,\alpha_n]$ and observe that $g|[0,\alpha_\omega]=f|[0,\alpha_\omega]$. Since $\alpha_g<\alpha_{n+1}$, we get $f\in B[g;\alpha_g]\subset U_g\in{\mathcal U}_F$.

Now it is necessary to adapt this proof to the general case of the space $C_k(\omega_1)$ of all (not necessarily integer-valued) functions on $\omega_1$.

For the subspace $C_k(\omega_1;{\mathbb Z})$ of $C_k(\omega_1)$ consisting of integer-valued functions, the proof of the Lindel"of property is relatively simple.

Given any open cover ${\mathcal U}$ of $C_k(\omega_1;{\mathbb Z})$, for every $f\in C_k(\omega_1;{\mathbb Z})$ find a countable ordinal $\alpha_f$ such that $f|[\alpha_f,\omega_1)$ is constant and the set $B[f;\alpha_f]=\{g\in C_k(\omega_1;{\mathbb Z}):g|[0,\alpha_f]=f|[0,\alpha_f]\}$ is contained in some neighborhood $U_f\in{\mathcal U}$ of $f$. For every ordinal $\alpha\in\omega_1$ consider the subspace $C_k(\alpha;{\mathbb Z})$ of $C_\omega(\omega_1;{\mathbb Z})$ consisting of functions that are constant on the interval $[\alpha,\omega_1)$. It is easy to see that the subspace $C_k(\alpha;{\mathbb Z})$ is Lindel"of (moreover, countable). Put $\alpha_0=0$ and construct an increasing sequence $(\alpha)_{n\in\omega}$ of countable ordinals as follows. Assume that a countable ordinal $\alpha_n$ has been constructed. Consider the open cover $\{B[f;\alpha_f]:f\in C_k(\alpha_n;{\mathbb Z})\}$ of the Lindel"of subspace $C_k(\alpha_n;{\mathbb Z})$ and find a countable subset $F_n\subset C_k(\alpha_1;{\mathbb Z})$ such that $C_k(\alpha_m,{\mathbb Z})\subset\bigcup_{f\in F_n}B[f_n;\alpha_{f_n}]$. Let $\alpha_{n+1}=\sup\{\alpha_f+1:f\in F_n\}$.

We claim that for the countable set $F=\bigcup_{n\in\omega}F_n$ the family ${\mathcal U}_F:=\{U_f:f\in F\}\subset {\mathcal U}$ is a required countable subcover of $C_k(\omega_1;{\mathbb Z})$. Given any function $f\in C_k(\omega_1;{\mathbb Z})$, use the continuity of $f$ at the ordinal $\alpha_\omega=\sup_{n\in\omega}\alpha_n$ and find $n\in\omega$ such that $f([\alpha_n,\alpha_\omega])=\{f(\alpha_\omega)\}$. Find a unique function $g\in C(\alpha_n;{\mathbb Z})$ such that $g|[0,\alpha_n]=f|[0,\alpha_n]$ and observe that $g|[0,\alpha_\omega]=f|[0,\alpha_\omega]$. Since $\alpha_g<\alpha_{n+1}$, we get $f\in B[g;\alpha_g]\subset U_g\in{\mathcal U}_F$.

Now it is necessary to adapt this proof to the general case of the space $C_k(\omega_1)$ of all (not necessarily integer-valued) functions on $\omega_1$.

For the subspace $C_k(\omega_1;{\mathbb Z})$ of $C_k(\omega_1)$ consisting of integer-valued functions, the proof of the Lindel"of property is relatively simple.

Given any open cover ${\mathcal U}$ of $C_k(\omega_1;{\mathbb Z})$, for every $f\in C_k(\omega_1;{\mathbb Z})$ find a countable ordinal $\alpha_f$ such that $f|[\alpha_f,\omega_1)$ is constant and the set $B[f;\alpha_f]=\{g\in C_k(\omega_1;{\mathbb Z}):g|[0,\alpha_f]=f|[0,\alpha_f]\}$ is contained in some neighborhood $U_f\in{\mathcal U}$ of $f$. For every ordinal $\alpha\in\omega_1$ consider the subspace $C_k(\alpha;{\mathbb Z})$ of $C_\omega(\omega_1;{\mathbb Z})$ consisting of functions that are constant on the interval $[\alpha,\omega_1)$. It is easy to see that the subspace $C_k(\alpha;{\mathbb Z})$ is Lindel"of (moreover, countable). Put $\alpha_0=0$ and construct an increasing sequence $(\alpha_n)_{n\in\omega}$ of countable ordinals as follows. Assume that a countable ordinal $\alpha_n$ has been constructed. Consider the open cover $\{B[f;\alpha_f]:f\in C_k(\alpha_n;{\mathbb Z})\}$ of the Lindel"of subspace $C_k(\alpha_n;{\mathbb Z})$ and find a countable subset $F_n\subset C_k(\alpha_n;{\mathbb Z})$ such that $C_k(\alpha_n,{\mathbb Z})\subset\bigcup_{f\in F_n}B[f_n;\alpha_{f_n}]$. Let $\alpha_{n+1}=\sup\{\alpha_f+1:f\in F_n\}$.

We claim that for the countable set $F=\bigcup_{n\in\omega}F_n$ the family ${\mathcal U}_F:=\{U_f:f\in F\}\subset {\mathcal U}$ is a required countable subcover of $C_k(\omega_1;{\mathbb Z})$. Given any function $f\in C_k(\omega_1;{\mathbb Z})$, use the continuity of $f$ at the ordinal $\alpha_\omega=\sup_{n\in\omega}\alpha_n$ and find $n\in\omega$ such that $f([\alpha_n,\alpha_\omega])=\{f(\alpha_\omega)\}$. Find a unique function $g\in C(\alpha_n;{\mathbb Z})$ such that $g|[0,\alpha_n]=f|[0,\alpha_n]$ and observe that $g|[0,\alpha_\omega]=f|[0,\alpha_\omega]$. Since $\alpha_g<\alpha_{n+1}$, we get $f\in B[g;\alpha_g]\subset U_g\in{\mathcal U}_F$.

Now it is necessary to adapt this proof to the general case of the space $C_k(\omega_1)$ of all (not necessarily integer-valued) functions on $\omega_1$.