From the Leray-Serre spectral sequence for a covering map $Y\to X$, which is a fibration with discrete fibers, we get an isomorphism $H^p(Y)\cong H^p(X,\mathcal H^0)$, where $\mathcal H^0$ denotes the local system of coefficients which at each point of $X$ has group equal to $H^0(p^{-1}(x))$.
If the covering is of $n$ sheets, then $\mathcal H^0$ is locally $\mathbb Z^n$, with the fundamental group of $X$ acting by permutation of the standard basis according to the monodromy permutation representation.
Now the local system $\mathcal H^0$ corresponds to a sheaf $\mathcal F$ on $X$, and for sensible $X$ (paracompact, say), one can compute singular cohomology with coefficients in the local system as sheaf cohomology with coefficients on the sheaf $\mathcal F$. If $X$ has a good finite cover $\mathcal U$ (in the sense of the book of Bott-Tu) then one can compute sheaf cohomoogy $H^p(X,\mathcal H^0)$ as the Cech cohomology $H^p(\mathcal U,\mathcal H^0)$. Looking at the complex which computes this by definition, we see that the Euler characteristic of $H^p(\mathcal U,\mathcal H^0)$, and therefore of $H^\bullet(X,\mathcal H^0)$, is $n$ times that of $H^\bullet(X,\mathbb Z)$. Notice that the existence of good covers implies being of bounded finite type, as you say (but I think it even implies that the space of of the homotopy type of a CW-complex, namely the nerve of the good covering... so all this might not get us much)
The fact that the Euler characteristic of a sensible space with coefficients on a local system of coefficients which locally looks like $\mathbb Z^n$ is $n$ times that of the space should be written down somewhere, but I cannot find it now.
There is this answer by Matt but he does not give a reference.
Your argument relating $g$ to $\chi$ is wrong. It's actually $\chi = 2 - 2g$. Check this for a torus: one zero cell, two one-cells, one two-cell: $\chi = 0$. But $g = 1$.
It looks as if you thought that the zero-cell and the 2-cell cancelled, but they should actually both be counted as $+1$.
Best Answer
Given an $m$-dimensional CW-complex $X$, one can lift the CW-structure to a CW-structure on $Y$ by lifting the characteristic maps $\varphi_\alpha : D^k \to X$ to the cover $p : Y \to X$, which can be done since $\pi_1(D^k) \cong 0$.
If degree of $p : Y \to X$ is $n$, there are exactly $n$ lifts of $\varphi_\alpha$ to $Y$. So for each $k$-cell $e^k$ in $X$, there exists $n$ $k$-cells in the lifted CW-structure on $Y$ which are mapped homeomorphically onto $e^k$.
Let $C_i$ be the number of $i$-cells in $Y$, and $C'_i$ be the number of $i$-cells in $X$. From the above analysis, we derive that $C_i = n \cdot C_i'$ for all $0 \leq i \leq m$. Using the fact that Euler characteristic of a CW-complex $X$, namely the alternating sum of it's betti numbers, is the same as alternating sum of it's number of cells (dimension of the cellular cochain groups), we conclude
$$\chi(Y) = \sum_{i = 0}^m (-1)^i C_i = \sum_{i = 0}^m (-1)^i n C'_i = n \chi(X)$$
as desired $\blacksquare$
I missed the essential question of OP up there. If $p : Y \to X$ is a covering map, $A \subset X$ is a subspace then $p|_{p^{-1}(A)} : p^{-1}(A) \to A$ is also a covering map.
In particular, take $A = e^k$ where $e^k$ is one of the cells in $X$. As the only covering space of disks are trivial (of the form $e^k \times D$ where $D$ is a discrete set), and $p$ is of degree $n$, $e^k$ must lift to $n$-many $k$-cells in $Y$.