Because the A-1 was among the first cameras to be able to communicate (primitive) informations from/to the flash unit (IIRC flash ready, set sync speed and get the aperture set on the flash when using compatible units like the 199A) and thereby has more contacts than the single flash trigger one, beware what signals the ST-E3 could send to the camera if mounted in the hot shoe and contacts match. Otherwise, it should be safer to connect it to the PC sync port via an adapter - then using manual mode it should work without issues.
When buying an A-1, check for the mirror foam, and the shutter "squeak" issue. They could need maintenance.
Thanks alot for your answer. If it proves to be a hassle, I will either get a dedicted flash, or simply use the camera without flash. I get it with a FD 35 F2.0 and a 100 F2.8. I will check for those things you mention.