Primes less than sqrt(1033) for which I know of no really obvious tricks (i.e. the digit adding tricks for them aren't so simple one can trivially do them in your head): 7, 13, 17, 19, 23, 29, 31

Also, since the scenario said we've done this many times and we haven't been trolleyed yet, it can't be all that easy to get trolleyed.

(eta: why the -1? Both points seem solid to me, even in the light of the additional trick below - there are several possible factors remaining, and it's not like I was enumerating these during my 2 minutes. Moreover, the 1001 trick works for 1033, but doesn't help so much with other typical numbers of that general magnitude - say, 1537, that it's something you're liable to do by accident)