Alternately, UBI allows you to quit your job, move to WV and pay $200 for rent.
What constrains your landlord from jacking up rent is competition. If he jacks up rent by $200 but the landlord across the street has an empty apartment he'd be glad to rent out for $1000 he'll think twice about jacking your rent.
The housing crisis in America is a supply issue. UBI has nothing to do with it.
Inflation is primarily a monetary phenomenon. If they pay for UBI with deficit spending it might crank up inflation, but if it's fully funded it won't.