Addition and multiplication behave well under mod; division does not, except by numbers coprime to the modulus.
Interview uses: last-digit questions (work mod 10), divisibility rules, cyclic processes, and clock problems.
Fermat's little theorem is worth knowing: a^(p-1) ≡ 1 mod p for prime p not dividing a. It makes questions like "last digit of 7^100" immediate rather than tedious.
The practical shortcut in mental arithmetic is casting out nines - working mod 9 via digit sums - which catches most arithmetic slips in a couple of seconds.