Problem source: 2c on the UMD January, 2018 topology qualifying exam, seen here https://www-math.umd.edu/images/pdfs/quals/Topology/Topology-January-2018.pdf I have an argument for it, but I am not at all sure that it is correct. In particular, I would like to know 1) if it is correct and 2) if there is a better way of doing it (especially if my way is incorrect).
Let X be a connected CW complex with $\pi_1(X)$ finite. Let Y be a CW complex. Let $f:X\to Y\times S^1$ be a continuous map. Show there exists a $g:X\to Y$ such that $f$ is homotopic to the map $(g,1):X\to Y\times S^1$.
I am fairly confident in the first part of the proof, ie that $|f_*\pi_1(X)|<\infty$ (although my group theory is rusty). Since $f$ is continuous, it induces a group homomorphism on $\pi_1(X)$, so 1) $f_*\pi_1(X)\simeq\pi_1(X)/Ker(f_*)$ and 2) $f_*\pi_1(X)\leq\pi_1(Y\times S^1)\simeq\pi_1(Y)\times \mathbb{Z}$. In particular, (1) gives that $f_*\pi_1(X)$ is finite, hence $\simeq H\times \{0\}$, where $H$ is some subgroup of $\pi_1(Y)$, by (2).
The part I am much less sure of (that it is either correct or necessary) is the remainder. Take $p:Y\times\mathbb{R}\to Y\times S^1$ by $p(y,x)=(y,e^{2\pi ix})$. This is a covering map, and $p_*\pi_1(Y\times\mathbb{R})=\pi_1(Y)\times\{0\}$, so $f$ lifts to some $\tilde{f}$. Since the image of any closed loop in $X$ is taken to a closed loop that does not wind around $S^1$ (else, we would get a generator for an infinite fundamental group), $\tilde{f}$ is homotopic to some $(g,0)$, which maps to $(g,1)$ under $p$.
I have not used the fact that X and Y are CW complexes anywhere, but the only theorem that I know offhand in the context of CW complexes, subsets, and homotopy of functions would be the Homotopy Extension Theorem, which wouldn't make sense here (since $f(X)$ is not necessarily a CW complex).
This is basically correct and is the natural argument to make but your argument at the end is unclear. The reason that $\tilde{f}$ is homotopic to some $(g,0)$ is simply that $\mathbb{R}$ is contractible, so you can take $\tilde{f}$ and contract its second coordinate to $0$ via a homotopy. Your explanation seemed to be mixing this up with the reason that the lift $\tilde{f}$ exists at all.
As for where the assumption that the spaces are CW-complexes comes in, you need something about $X$ to be able to say that the lift $\tilde{f}$ exists at all (the usual lifting theorems for covering spaces only work for sufficiently nice spaces). In particular, it suffices to know that $X$ is path-connected and locally path-connected, which follows if you know that $X$ is a connected CW-complex. The assumption that $Y$ is a CW-complex is completely unnecessary.
By the way, the fact that $f_*\pi_1(X)$ is finite is completely trivial and requires no group theory: it is the image of the finite set $\pi_1(X)$ under a function.