How many values of x between 0.01 and 1 does the graph sin(1/x) cross the x-axis?

$31$ values.
Given a domain of $\left[0.01 , 1\right]$, $\frac{1}{x}$ has a range of $\left[1 , 100\right]$. Thus, the question is equivalent to asking how many times $\sin \left(x\right)$ crosses the $x$-axis on the interval $\left[1 , 100\right]$.
As the graph of a function crosses the $x$-axis at the points where the function evaluates to $0$, and $\sin \left(x\right) = 0 \iff x = n \pi , n \in \mathbb{Z}$, all that remains is to count the multiples of $\pi$ in the interval $\left[1 , 100\right]$
We can verify that $0 < 1 < \pi$ and $31 \pi < 100 < 32 \pi$, meaning the only integers in which $n \pi \in \left[1 , 100\right]$ holds are $n = 1 , 2 , \ldots , 31$. As there are $31$ such values, $\sin \left(x\right) = 0$ has $31$ solutions on $\left[1 , 100\right]$. As this is equivalent to our original problem, we have that $\sin \left(\frac{1}{x}\right)$ crosses the $x$-axis $31$ times for $x \in \left[0.01 , 1\right]$