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How should I approach this problem?
I know how to do 1. but I am not sure how to approach 2.
This is not really an astronomy so much a basic Maths problem.
Put the Earth at the origin and the star at $\vec{r}(t)$ with a position $\vec{r_0}$ at $t=0$. Define your axes such that the star is in the z direction and its tangential motion components are along the x and y axes.
$$\vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k},$$ where these components are the two proper motion velocities and the radial velocity.
Assume no acceleration, so that $$\vec{r} = \vec{r_0} + \vec{v} t $$
Use Pythagoras to get the distance to the Earth using the components of $\vec{r}$ at any time $t$ and then find the time at which this is a minimum (and if you are unsure how to do that $\rightarrow$ Maths SE).
