What is the most pressing problem in Act 4, in Hamlet,?

Answer 1

Only one? There are any number of them.

Hamlet's old "friends," Rosencrantz and Guildenstern, have been brought in, essentially to spy on Hamlet. Hamlet knows it, and doesn't trust them.

Claudius knows that Hamlet is on to him, and contrives to have Hamlet shipped out to England. What Hamlet doesn't know is that Claudius is sending along letters to the English King indicating that Hamlet is to be executed upon his arrival there.

Ophelia has gone insane. Hamlet murdered her father, Polonius, thinking that Polonius (hiding behind a curtain in Gertrude's room) was Claudius. Ophelia's brother, Laertes, returns from France, itching for revenge.

And, as if all that isn't enough, Fortinbras and his army are on the march across Denmark.

Those are all pressing problems. Take your pick.