Here is the way you get a non-vanishing holomorphic section which will trivialize your line bundle.

Denote $U_1=\{[x:y]; x\neq 0 \}$ and $U_2=\{[x:y]; y\neq 0\}$, this is a covering of $\mathbb P^1$. The transition function of the bundle is $g_{12}([x:y])=e^{y/x}$ (you could also take $e^{x/y}$, this wouldn't change the argument), defined on $U_1 \cap U_2$.

Then you may defined the following holomorphic functions: on $U_1$, you put $s_1([x:y]):=e^{y/x}$ and on $U_2$, $s_2([x:y]):=1$. Thus $s_1$ and $s_2$ are non-vanishing holomorphic functions, and on $U_1 \cap U_2$, you have $s_1=g_{12}s_2$ so that they form a section of our line bundle.

EDIT: In particular the degree of the line bundle is 0! 

Moreover, on a Riemann surface (or more generally on any smooth projective complex variety), any line bundle admits a meromorphic section, so that the correspondence your are talking about still holds.