Every fingering that fits a hand
3544
moderate
3141
hard · no 5
3144
hard · no 5
314x
easy · no 5
xa9b
easy · no 5
What is true of it
A♯m7 is A♯, C♯, E♯ and G♯ — 1, b3, 5 and b7 counted from A♯.
These are also the notes of C♯6/A♯ and D♭6/A♯. Which name is right depends on what is underneath and what came before — the notes alone do not decide.
On an instrument that cannot reach all 4, the 5 goes first, in that order.
The easiest of the 5 fingerings below is 3544: 4 fingers, held at the 3rd fret.
None of them uses an open string, so this chord moves up the neck unchanged — the same shape 1 fret higher is the chord a semitone above.
Where its notes are on the neck
The brass dots are A♯, the root. Everything else is a chord tone with its degree printed in it.
With a different note underneath
A♯m7/C♯
6868
A♯m7/E♯
aa9b
A♯m7/G♯
1111
An inversion is the same chord with a different note lowest. The name changes to a slash name because the bass is what a listener hears as the foundation.