Do I need a screen protector for a MBA 13?

There is no need to use any of these protectors.

Keep the liquids away from the computer.

https://support.apple.com/en-us/HT204172

Don't use cases, shells, covers or "skins". All of them impede proper heat dissipation. They also trap abrasive contaminants between the case and shell, probably causing more wear than would otherwise occur. Never place anything over the keyboard when operating the computer. Read Mac notebooks: Operating temperature - Apple Support.

Purchase a sleeve to keep the Mac protected when it is not in use.

I have to respectfully disagree with the "don't use a shell" advice.

I have been using shells on my MacBook, 2 MacBook Pros and 4 different Macbook Airs, for several years (since unibody MacBooks were introduced). Not only do they function to give me a far better grip for carrying, they have saved my Air cases, at least twice. Once when my two year old made off with my Air and dropped it on a tile floor. The corner of the shell shattered, and dissipated the drop force, leaving no damage to the actual computer. The other time, my 11" Air when flying off a coffee table after someone tripped over the (mag"safe") cord. Also the computer was saved from corner dent damage. While these are anecdotal for sure, in hindsight, the insurance was well worth it. May I mention again, that a good shell makes carrying the computer far safer ("gription").

I used to think that the shell would trap abrasive dirt, so I used a perforated inCase cover, under the theory that the perforations would "release" any grit. I have bought all my shells at the Apple Store (Do you think they would sell them, if there was a heat problem?), but I digress. Those covers were not the greatest, because they would hold up well to periodic removal (for checking, cleaning, system servicing). Maybe that is why inCase later discontinued them. But I never had a single scratch during the few years I used them.

Later, I switched to full cover cases (inCase hammered finish). They do have a few strategically placed holed sections (I suppose for cooling). These stand up well to periodic removal. My paranoia more than anything else (removing every 6-8 months, or so, and once when I had to get a new battery for an Air). ZERO scratches...I mean a perfect finish, no evidence that the cover is doing anything whatsoever, except maybe actually protecting it. In fact, the Genius Bar Tech who checked in my Air for battery service remarked at the perfect exterior condition of a 3 year old Air. Moreover, my temps run completely normal, even on games (Portal 2, Call of Duty, various Lucas-Games). BTW, the Air cools though a rear fan vent, and not much through the case itself (except for a perforated location running alon the underside of the logic board), which the shell manufacturer provides for.

I am firmly convinced that it is FUD (fear, uncertainty and doubt) that leads to the theory that shells are bad (I was there once). I suppose there may be cheap, poorly designed shells that could be a problem. But I guess that's why I always used only those sold by Apple themselves. (mostly inCase, and one Speck).

The number one reason I have used shells is for the grip, but in the end it has proven to me to be protective as well. I don't think in the future I will be removing them as often (for OCD reasons), and will only remove for case-open servicing.

BTW, I do use a sleeve, too.

To specifically address the OP,..

You don't specify the make or model shell you have chosen. Not for a moment may I presume that all shells are designed properly, so your particular choice can merit no definitive opinion from anyone else, with so limited information given. As for the "screen protector", I see no reason for it.