Same cell structure have isomorphic fundamental groups

Let $$X$$ and $$Y$$ be two topological spaces both with the following cell structure

2 0-cells

4 1-cells

3 2-cells

1 3-cell

Can I conclude anything about the fundamental groups of $$X$$ and $$Y$$? Are they isomorphic?

No, they don't have to. You can collapse each boundary of each cell to a single point to obtain the wedge sum of all cells (together with a single isolated $$0$$-cell), which are spheres of different dimension. This will give you $$\mathbb{Z}*\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$$ as the fundamental group on one component. The other option is to make a disk out of two $$1$$-cells and one $$2$$-cell and again wedge product of the rest. Since we now have only two $$1$$-cells left then the fundamental group is $$\mathbb{Z}*\mathbb{Z}$$.
These examples have an isolated $$0$$-cell but this can be fixed again by using one of the $$1$$-cells to connect it to the rest of the body (so it's like a hair growing out of the center of the wedge sum). That way we reduce the number of $$1$$-cells in both cases by $$1$$ obtaining fundamental groups of $$\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$$ and $$\mathbb{Z}$$ respectively.
By making all $$1$$-cells glued with all $$2$$-cells (i.e. the 2 dimensional disk) and the sole $$3$$-cell added via wedge sum (i.e. $$D^2\vee S^3$$) you obtain a cw complex with trivial fundamental group. So as you can see lots of choices...