If you were to use 'Now', it would have to be 'Now that'.
'Prospective' can be replaced with 'future' (Though it doesnt have exactly the same meaning), which means that he's not yet the tenant of the new apartment.
Also, 'will be able to (...) move in" indicates that he isn't able to move in yet, but he will be in the future.
Using 'Now' instead of 'Once', the sentence would look like this:
Now that the lease is approved, the new tenant is able to officially move in the new apartment.