To “bowl over” has two different meanings. It’s literal meaning is to knock or push someone over. So if someone runs into you and you fall, you’ve been “bowled over.” The other meaning is a figurative meaning. It’s when you feel shocked or overwhelmed. Almost like the same feeling you would have if someone really did bowl you over. So if you are very surprised about something or someone you can be “bowled over.”  So basically your sentence means “I was completely surprised but somehow I managed to act cool and agreed to meet him.” Hope this helps!

It means "extremely surprised". So surprised that you could fall over.