I would plant a hedge of clumping bamboo all along the fence. Something like Bambusa oldhamii or Otatea acuminata sap. aztecorum. That's quite an unfortunate view, I doubt a vine on the fence would be enough to the sight of both levels of the building. Good luck.

Cheap, reliable, reasonably fast (water for good growth) = Ligustrum japonicum, AKA Privet, available at any big box store, Plant them about 3' apart and at least 30" away from the fence itself. Give it room to get dense and solid: do not plant it right next to the fence.

Now, for INSTANT help until the Ligustrum grow in, you can get opaque fabric to attach to your fence to make the view of the other side go away TODAY (or as soon as the order comes). Google up "privacy fence fabric". It's fairly inexpensive and well worth the money.

I agree with Hoovb. I would take one additional approach depending on funds, and purchase a tree to block the view of those living in the second story apartments.

I like Hopseed Bush / dodonea viscosa for fast growth and agree about at least one tree planted closer in, away from the fenceline for additional privacy from that second story overlooking your yard.

I lived in a similar situation. First as large of a patio cover as possible to block out the view down with a tall trellis around it with bougainvillea helped a lot since most time spent out doors was on the patio. Maybe a lath house?

That plus the fence fabric screen or those slats that fit in the chain link would be a quick and huge improvement...if you already have a patio cover I would definitely close the sides or at least the side facing that apartment in with plants or perhaps some other material.