Bishop fired home from the penalty spot in the 21st minute after Phil Edwards had chopped down Nicky Adams.
Then, on the stroke of half-time, the 26-year-old burst through and finished superbly past Kenny Arthur for the mid-table side.
Stanley, safe from relegation, had chances in the first half, but Bury keeper Jim Provett did well to thwart Paul Mullin and Bobby Grant.
The home side dominated after the break but were wasteful.
Andy Todd blasted over in a good position, substitute Leighton McGivern scuffed a golden opportunity and Mullin was denied by Provett from close range to end a disappointing season for Accrington.