What are some problems an actor might have when trying to play Hamlet on stage?

It's a mive line load. Your ability to scan all those lines for their meter, understand every word and sentence, memorize them and perform well is a big project. Also, he is rarely offstage, so it is physically demanding. This is about the biggest and most difficult role an actor can take on. You would need to be focused and extremely disciplined so that you were taking care of your mind, body and spirit during the entire rehearsal and performance process. Working out, getting plenty of sleep, watching your diet carefully, steering away from junk food,alcohol etc. These things would be necessary to allow you to maintain the creative energy needed to pull off this part.