Try a couple of pots of morning glory seeds. They’re cheap, grow fast and thick and are fun to look at in the morning. See if you can find any that grow up the trellis left handed instead of right handed.
Depending on the surface above you, you might also be able to mount a rolling shade of some sort; it might be less obtrusive and then you could simply roll it down when you were looking for privacy.
Check your apartment rules before you make a big investment. Many apartment communities limit what can be put on the patio because they want the area to look clean and consistent. I would also take my clues from what has been allowed for other residents. Putting something up that draws attention might cause friction with your landlords.