I am pretty sure that the answer to this question is unknown. And I would guess it should be hard (via impossible) to construct such a manifold. Here is some argumentation:

In dimension $6$ the classification of symplectic manifolds with same homology as $\mathbb CP^3$ admitting a Hamiltonian $S^1$ action with isolated fixed points was done by McDuff http://arxiv.org/abs/0808.3549 .

After the classification is done McDuff notices that all the (four) examples are in fact three-dimensional algebraic Fano varieties (i.e., the symplectic structure comes from a Kahler form).

On the other hand, as far as I understand for the moment (as ridiculous as it sounds) there is no single example of a symplectic manifold admitting a Hamiltonian $S^1$ action with isolated fixed points, that is known to be non-algebraic (see for example: Hamiltonian $S^1$ actions with isolated fixed points)

If we would now indeed try to look for an 8 dimensional example different from $\mathbb CP^4$ that is additionally algebraic, we would need to look for such a Fano four fold. BUT, here the situation is as follows: in all even dimensions $\mathbb CP^{2n}$ are the only known (as for today) Fano varieties with $H^{2k}=\mathbb Z$, $H^{2k-1}=0$. It is true though, that $4$-dimensional Fano varieties are not yet classified, contrary to $3$-dimensional ones.

If on the other hand by any miracle we will find some non-algebraic example, this will answer to the following question, which I am sure is still open at the moment: Compact Symplectic Fano (strongly monotone) manfiolds

I guess the surest way to answer this question will be to try to see if McDuff's classification can be generalised to this dimension...

I am pretty sure that the answer to this question is unknown. And I would guess it should be hard (via impossible) to construct such a manifold. Here is some argumentation:

In dimension $6$ the classification of symplectic manifolds with same homology as $\mathbb CP^3$ admitting a Hamiltonian $S^1$ action with isolated fixed points was done by McDuff http://arxiv.org/abs/0808.3549 .

After the classification is done McDuff notices that all the (four) examples are in fact three-dimensional algebraic Fano varieties (i.e., the symplectic structure comes from a Kahler form).

On the other hand, as far as I understand for the moment (as ridiculous as it sounds) there is no single example of a symplectic manifold admitting a Hamiltonian $S^1$ action with isolated fixed points, that is known to be non-algebraic (see for example: Hamiltonian $S^1$ actions with isolated fixed points)

If we would now indeed try to look for an 8 dimensional example different from $\mathbb CP^4$ that is additionally algebraic, we would need to look for such a Fano four fold. BUT, here the situation is as follows: in all even dimensions $\mathbb CP^{2n}$ are the only known (as for today) Fano varieties with $H^{2k}=\mathbb Z$, $H^{2k-1}=0$. It is true though, that $4$-dimensional Fano varieties are not yet classified, contrary to $3$-dimensional ones.

If on the other hand by any miracle we will find some non-algebraic example, this will answer to the following question, which I am sure is still open at the moment: Compact Symplectic Fano (strongly monotone) manfiolds

I guess the surest way to answer this question will be to try to see if McDuff's classification can be generalised to this dimension...