Density of Ni vs Cu using unit cell
I think you need to make at least one more assumption along the way to determining the relative densities. But let’s take it stepwise:


We are given a primitive cubic cell for nickel and a face-centred cube for copper.
We therefore conclude that per unit cell one atom of nickel and three atoms of copper are present.
We know from the periodic table that copper is a bit heavier than nickel ($63.5$ versus $58.7$)
Therefore, it is clear that copper’s unit cell is more than four times as heavy as nickel’s. But we don’t know their sizes.
From mathematical considerations we can also calculate how much of each unit cell is filled with atoms ($74~\%$ for fcc; somewhat less which I am too lazy to calculate now for primitive cubic) — this again means that there should be more weight per volume in the copper unit cell.
What seems to be lacking in my eyes is a way to compare the unit cell sizes, whether by atom size or unit cell dimensions. However, since nickel and copper are right next to each other in the periodic table, it may be safe to assume their sizes to be almost identical.
If that assumption is safe, we can use either route to clearly show how copper’s density is higher. But note that if copper’s atoms were significantly larger than nickel’s, this reasoning would break down extremely quickly.