Neither of my kids are burpers, from breast or bottle. They don't necessarily need to burp.
It's possible his fussiness could come from being gassy, though. If it's something you would want to do you could try getting him to the breast. Skipping the bottle would eliminate the risk of air getting in. He's definitely young enough to learn to latch if there are no other issues.
Aside from that there are several different positions to burp a baby in. I always set the baby on my lap sitting up with their chin supported in my hand and my arm going down their chest. Then you can use your other hand to pat their back. That seems to work well. You can also try it with the baby laying flat on your lap or your arm.
In my experience most babies tend to have a fussy time of day when nothing will console them. Commonly it's in the early evening, but all babies are different. Good luck!