People have flown Tiger Moths from The UK to Australia. There's no reason that, if you choose the right aircraft, you shouldn't be able to do a similar trip. A C172, for example, would be more than capable of handling a new PPL's hour building (assuming he was not a complete incompetent) and then doing such a trip. After all, it's nothing more than 100 or so hours of hour building, but in a straight line.
As for shipping an airplane from the States; again, something thats often done. You can either have it ferry-flown, or put into a container and shipped. Both options are likely to be expensive.
When I was a PPL with 150 hours I flew a C172 from coast to coast across the US. This is obviously rather a bigger trip that you're planning, but it goes to show that it is entirely achievable. I say go for it!
A lot of people fly via Egypt on such a trip. See my website for a writeup of flying Egypt to the UK
katamarino.co.uk - Adventure Flying Resources