Green’s Dictionary of Slang

1. to give up trying.

1938

D. Runyon ‘Pick the Winner’ in Runyon on Broadway (1954) 318: ‘It looks as if I will have to blast, [...] the professor is a nice guy, but,’ he says, ‘he does not loosen so easy.’.

2. to move fast.

1973		Eble Campus Sl. Nov. 1: blast – to hurry; to take something quickly.
2003		A.N. LeBlanc Random Family 29: Cesar blasted out of a nearby pool hall.