Hi Agame! These two questions do not have the same meaning. The first one suggests that there is a problem and therefore it is difficult for you to buy the person a beer. For example, you could finish the first question like this: How am I supposed to buy you a beer when I don’t have any money? or How am I supposed to buy you a beer when you are never in the pub at the same time as I am? The second question suggests that you are not sure whether or not it is ok. Perhaps the other person is going to leave the pub but you are not sure about this. You’d like to buy one but you think the other person might leave or not want a beer (for some reason or other) so buying one could be pointless. Hope this helps! Warmest wishes. Cate 😊

Well, w.r.t the second sentence, I don't think anyone would want to buy a bear!! 😉. But a beer, yes. The first sentence has already been explained by others.

Hi Agame, Cate's answer is perfect. To add some more context here is an example dialogue: ⭐️EXAMPLE DIALOGUE ⭐️ - Mate, can you buy the next round? I'm skint! - How am I supposed to buy you a beer? I don't have a penny to my name right now. - Oh come on, I bought you a beer last week! - Fine, you're really twisting my arm here. - Thanks, you're the best! ⭐️We can also use it in many other contexts: - How am I supposed to do that? - How am I supposed to climb up there?! I'm far too unfit. - How am I supposed to open this tin, we left the tin-opener at home! - How am I supposed to follow along with the dance instructor? I've got two left feet! It's a great collocation to say that we are confused or unsure about how we are going to complete an action.

"supposed" The meaning cannot be determined without context because, as you will see by consulting a dictionary, the verb "suppose" has several meanings.