If you updated to a AM3+ motherboard/CPU and are still using a X1300, then that is the problem for running Skyrim, since it's a budget card released a decade ago.
If it is still a Pentium 4 and you think the advice you were given to upgrade the system was mistaken- and that only the Graphics card is the problem, then you were mistaken, since a Pentium 4 would be just as much an issue for running that game as a X1300.