Talk to your new landlord. Explain the situation. If necessary, provide
veterinary records that prove the neuter, prove he's current on all his
vaccinations. Provide references from your current residence, if at all
possible. If you're renting now, ask your current landlord to talk to your
new landlord. If your cat was well behaved and didn't rip up furniture or
carpeting or do damage to the apartment, a great many landlords may be
swayed by not only having that kind of information, but also that their new
tenant is devoted to their pet and will go those extra steps to keep the
animal with them. Keep in mind it's possible that your new landlord will ask
for a declawed cat, but many will allow claws if you explain about Soft
Paws, or if you have those references to show that your kitty is not a
door-shredder.

If worse comes to worst and you have no alternative to this move and to
giving up kitty, shelters have a pretty good track record for finding great
homes for good natured cats.

Have you exhausted all possible alternatives to the place you have chosen?
I had cats when I was renting, and I found that it was often difficult to
rent the places I would have preferred (both price and location).  However,
my cats (one cat at a time at that time, like you) were my babies and I
absolutely would not move somewhere that would not accept cats.  Sometimes I
was able to reach an agreement with a landlord by offering to make an extra
security deposit and could also offer references from previous locations.
You could even offer to let the new landlord inspect your current premises
to see that there is no damage.  If that won't work, please keep looking ...
there surely must be some place in your price range where you can take your
cat.  You might even want to consider sharing an apartment with another
cat-lover to reduce costs.

MaryL