What problem did Robert Frost face? How did he face it?

I am assuming that you mean to ask what problem he is facing in the poem "Road Not Taken."

Looking at the poem literally, he is faced with the problem of which road to take when he comes to a fork in the road.  He tries to figure out which road to take but can't really decide if/how they are different.

He doesn't actually tell us which one he takes.  He says (last stanza) that he will say he picked the road less traveled (when he looks back on this) but it is not clear that he really did.  And in fact it's not clear that one road WAS less traveled.

So if I had to answer the second question I'd say he faced it by taking one road and then claiming that the road he chose was special.