Let $Y$ be a $n$-dimensional $CW$ complex. This means that $Y$ is a cellulation of topological spaces $$Y=Y_n\supseteq Y_{n-1}\supseteq\dots\supseteq Y_0$$ Where $Y_0$ is a non-empty set of points and $Y_k$ is obtained from $Y_{k-1}$ by adding a finite number, zero included, of $k$-cells. We call $\nu_k$ the number of $k$-cells attached to $Y_{k-1}$ and define the Euler-Poincaré characteristic $$\chi(Y)=\sum_{k=0}^n(-1)^k\nu_k$$ Given the 2-cell $D^2$ and a continuous function $f:S^1\to Y$, I'm asked to compute the Euler-Poincarè characteristic of $$D^2\cup_fY$$ This question was asked in a course between general topology and algebraic topology, so I have all the tools from general topology but I cannot use like (co)homology theories nor sequences.
I think I should try to prove this formula $$\chi(D^2\cup_f Y)=1+\chi(Y)$$ Which is somehow supported by some examples like \begin{align} 0=\chi(T_1)&=\chi(D^2\cup_f(S^1\vee S^1))=1+\chi(S^1\vee S^1)=1+(-1) \\ 1=\chi(\mathbb{P}^2_{\mathbb{R}})&=\chi(D^2\cup_f\mathbb{P}^1_{\mathbb{R}})=1+\chi(\mathbb{P}^1_{\mathbb{R}})=1+0 \\ 2=\chi(S^2)&=\chi(D^2\cup_f\{p\})=1+\chi(\{p\})=1+1 \end{align}
Any idea to prove it rigorously?
The problem is ill-posed as stated: Either it is your instructor's fault or you copied a problem incorrectly. The (main) issue with the current formulation is that the space $X=D^2\cup_f Y$ need not even be homeomorphic to the total space (aka geometric realization) of a CW complex. To get an example, take $Y=S^2$ (say, with the standard CW complex structure). Take a surjective (continuous) map $f: S^1=\partial D^2\to Y$ (you can find one using a space-filling curve). Then $X=D^2\cup_f Y$ does not admit a structure of a CW complex, for otherwise there would be a 1-dimensional subset $Z\subset X$ (homeomorphic to a finite graph) such that $X\setminus Z$ is a 2-dimensional manifold. However, no point of $Y$ has a manifold neighborhood in $X$. Thus, the definition of the Euler characteristic given in your class cannot be applied to $X$. There are several ways around this problem, for instance, one verifies that $X$ is homotopy-equivalent to a CW complex, but that would require one to prove that the Euler characteristic defined as in your question is homotopy-invariant (as defined, it is not even obviously a topological invariant).
Note that even if your space $X$ happens to be homeomorphic to a CW complex, without knowing more, it is unclear how to relate that CW complex structure to the one given for $Y$. You are also left with the task of proving that $\chi$ is a topological invariant (is independent of the CW complex structure), which, to the best of my knowledge, is impossible without using the some algebraic topology (which you did not study yet).
Most likely, your instructor simply forgot to assume that $f: \partial D^2\to Y_1$ (or you did not copy this assumption). (Another forgotten assumption is that the CW complex structure for $X$ is the "natural one", where $Y$ is a subcomplex.) Then the proof of the formula $\chi(X)=\chi(Y) +1$ is a straightforward cell-counting.