(**Edited** to include the case when $p$ is odd. Hopefully there are no mistakes as I don't usually work with $p$ odd.)

This is true, assuming you mean degree 0 maps. We have $ku\simeq l \vee \Sigma^{2} l \vee \dots \vee \Sigma^{2(p-2)} l$ where $l$ is called the Adams summand (usually denoted $BP\langle 1 \rangle$). Notice when $p=2$ that $ku=l$. The cohomology of $l$ is $$H^*(l; \mathbb{F}_p) \cong \mathcal{A}//E(1) = \mathcal{A} \otimes_{E(1)} \mathbb{F}_p$$
where $E(1)$ is the exterior algebra generated by the Milnor primitives $Q_0$ and $Q_1$ in degrees 1 and $2p-1$, respectively. For example, if $p=2$, then $Q_0 = Sq^1$ and $Q_1 = Sq^2 Sq^1 + Sq^3$. Thus
$$H^*(ku) \cong \mathcal{A}//E(1) \oplus \Sigma^2 \mathcal{A}//E(1) \oplus \dots \oplus \Sigma^{2p-4} \mathcal{A}//E(1).$$

Now look at each factor: by change of rings,
$$\text{Hom}_{\mathcal{A}}(\Sigma^{2n} H^{*}(l;\mathbb{F}_p),\mathcal{A}) \cong \text{Hom}_{E(1)}(\Sigma^{2n} \mathbb{F}_p, \mathcal{A}|_{E(1)})$$ for $0\leq 2n \leq 2p-4$, where $\mathcal{A}|_{E(1)}$ means restriction to $E(1)$.

There are no nonzero $E(1)$-maps from $\mathbb{F}_p$ to $\mathcal{A}|_{E(1)}$ because the bottom class of $\mathcal{A}$ supports nontrivial operations (e.g. $Q_0$). This shows the group is 0 for $n=0$, which is the only case to consider if $p=2$. If $p$ is odd, then the odd primary Steenrod algebra is generated by the Bockstein $\beta$ in degree 1 and the reduced powers $P^i$ in degree $2i(p-1)$. The first $P^i$ is in degree $2p-2 > 2p-4$, hence the generator of $\Sigma^{2n} \mathbb{F}_2$ can never hit a nonzero class when $n>0$, so $\text{Hom}_{E(1)}(\Sigma^{2n} \mathbb{F}_p, \mathcal{A}|_{E(1)}) = 0$ for all $n$ with $0\leq 2n \leq 2p-4$.

As noted in other answers, this statement is not true if we look at suspensions of $ku$ instead, because e.g. there is a nontrivial map $\Sigma^4 \mathbb{F}_2 \to \mathcal{A}$ where the generator of $\Sigma^4 \mathbb{F}_2$ hits $Q_0Q_1$ (since $Q_0Q_1$ supports no operation when $\mathcal{A}$ is restricted to $E(1)$, there is no longer a contradiction here).