In a wider sense, it couldn't be "completely" captured since many people don't rent. But I suppose you meant it from the perspective of an individual tenant.

Rents are set by supply and demand. If we assume your hypothesis is correct then complete capture via rents would have to be a stable state.

I do not think that's the case. First of all, if rents were to go up by that much (relative to everything else) then supply would increase (more building, renting out of currently empty properties, subdividing, etc.). Furthermore, regions with lots of empty homes could not keep rents high because landlords compete against each other. And as a consequence, people in high-price areas would move away to cheaper areas.

In conclusion, it is likely that rents and other prices would go up, but it is almost inconceivable that it would lead to no additional purchasing power for the poorest at all.

You fund it with a Georgist style land value tax.

https://en.wikipedia.org/wiki/Georgism

When you tax something you penalize it - so in this case you'd penalize owning land. This would seem to concentrate land in the hands of the wealthy as far as I can see.

which is another way to state that, once construction costs are repaid, you cannot earn money from your apartment.

UBI implies that property cannot be used as a way to extract money from someone else work.

So you're saying UBI has a better chance of working if we took away private land ownership and the government was everybody's landlord?