Let $f(x) = x^2$, and define $\alpha$ as follows, find $100\int_{-1}^{100}f\ d\alpha$.

When $\alpha$ is constant, the Stieljes integral is $0$. Suppose $\alpha(x) = m$ everywhere on $[c,d]$. If $P=(\{x_i\}_{i=0}^n, \{t_i\}_{i=1}^n)$ is a partition of $[c,d]$, then the Riemann-Stieljes sum over $P$ for $\int_c^d f\,d\alpha$ is $$S(P,f) = \sum_{k=1}^n f(t_k)(\alpha(x_k) - \alpha(x_{k-1}))$$ But since $\alpha$ is constant, $\alpha(x_k) - \alpha(x_{k-1}) = 0$ for every $k$,so $S(P,f) = 0$, regardless of the partition $P$ or integrand $f$. Thus every function $f$ is integrable with respect to $\alpha$, and $$\int_c^d f(x)\,d\alpha(x) = 0$$

So in the case of your first problem,

$$\begin{align}\int_{-1}^{100}f(x)\,d\alpha(x) &= \lim_{\epsilon\to 0+}\int_{-1}^{2-\epsilon}f(x)\,d\alpha(x) + \int_{2-\epsilon}^{3}f(x)\,d\alpha(x) + \int_{3}^{100}f(x)\,d\alpha(x)\\ &= 0 + \int_{2-}^{3}f(x)\,d\alpha(x) + 0\end{align}$$ Where the $2-$ indicates that you still have to account for the discontinuity of $\alpha$ at $2$ within the remaining integral.

In the second problem, they are asking you to calculate $$\int_0^n x\,d\alpha(x)$$ The function that needs to be integrated is whatever is between the $\int$ and the $d\alpha(x)$ (as used here - there is another notational convention where the integrand is given after the $d\alpha(x)$ and you have to guess what is included in it and what is not). The integrand does not have to named "$f$". And the function with respect to which you are integrating does not have to named "$\alpha$", though they still did so here.

So you integrate the function $x$ with respect to the function $\lfloor x \rfloor$.