The 21-year-old was without a club after leaving Hull City in the summer, bringing an end to a two-year spell with the Tigers.
He now joins a Robins side sitting seventh in the National League table as they look to maintain their challenge for promotion.
Brown has already gained experience at the level, having made 11 appearances for Hartlepool United during a loan spell last season.
That experience should help him settle into life at Altrincham, where he will be looking to turn his potential into a regular contribution in the first team.
The forward's arrival also gives the club another option in attack as the campaign progresses.
Altrincham have not disclosed the length of Brown's contract.