Write the multiplication table to find the inverses by brute force, or use the division algorithm.
You can try little arithmetic tricks: the inverse of 1 is 1, the inverse of 2 is 9, the inverse of 3 is 6, the inverse of 4 = 2^2 is 4^(-1) = 2^(-2) = (2^(-1))^2 = 9^2 = 13 (everything here's done modulo 17, of course),
the inverse of 5 is 7, the inverse of 10 = 2*5 is 10^(-1) = 2^(-1)*5^(-1) = 9*7 = 12 , etc.
If your other option is write down multiplication tables I think the above can be shorter.
Tonio