When modulating by a half step, one trick is to exploit the enharmonic equivalence between the German augmented-sixth chord and the dominant-seventh chord.
In A♯ minor, the German augmented-sixth chord is F♯ A♯ C♯ D𝄪; since this chord sounds the same as the V7 in B (F♯ A♯ C♯ E), you can easily resolve to B minor and strengthen the modulation from there.
Modulation and key change generally speaking uses the V chord of the new key. So you'd have a bar or two of F♯ or F♯7 leading into the new key of Bm.
You could also lead into that F♯ with an A♯diminished chord, using it as a pivot note. That gets the job done in one or two bars.
One thing I would like to add to the (excellent and quite complete) answers above.
There is a fundamental difference between arranging a pop song, and writing classical music (which generally lasts a lot longer) in terms of what you are trying to achieve.
For most pop music, you are trying to get to the listener by a sequence of events:
Hopefully the ...