A quest (the wild boar hunt one) requires me to give a beer to somebody.

I have no idea how to do that. If I just try to get a glass of beer from the barrel, my sim immediately starts drinking it and I have no way to store it. I can fill in a jug and then store that in my inventory, but that doesn't let me give out any beer either.

Help please, I'm stuck because of this...

Click on the barrel, there must be an option serve; and then you have the option to pick someone in the room

The man in question isn't in the room (and he won't come after being invited either). Do I have to become friends with him so that he'll accept my invitation then, or is there another way?

maybe you could go to where he is, and then buy a barrel and place it right there? would that be an option?
otherwise, yeah, try being friends with him before inviting him

There is a barrel near the campground in the woods...where you have to talk to the others on the hunt...first you click on the barrel and make the ale...afterwards you click again and "serve"...you will get a pop up of the people available...you just have to make sure the person you need is there...