Solution:

The required values of $n$ are all and only the multiples of 3.

Necessary condition Let $A$ be a multiple of 2022 whose decimal expression contains the digit 7 exactly $n$ times, and possibly other digits 0. Since 2022 is a multiple of 3, $A$ must also be a multiple of 3, and therefore (by the divisibility criterion for 3) the sum of the digits of $A$ must in turn be a multiple of 3. Now the sum of the digits of $A$ is $7n$, and this is a multiple of 3 only if $n$ is a multiple of 3.

Sufficient condition We observe that the number $70770 = 2022 \cdot 35$ is a multiple of 2022 and its decimal expression contains three digits 7 and two digits 0. This shows in particular that $n=3$ has the required property.

More generally, for every positive integer $k$ we observe that the number written by repeating $k$ times the block of five digits 70770, that is, the number
$$\underbrace{7077070770 \ldots 70770}_{k \text{ volte}} = 70770 \cdot \left(1 + 10^{5} + 10^{10} + \ldots + 10^{5(k-1)}\right)$$
is in turn a multiple of 2022 and its decimal expression contains the digit 7 $3k$ times and the digit 0 $2k$ times. This shows that every multiple of 3 has the required property.