If there is a lot of dust, then it surely will overheat. And if it overheated with Oblivion and Fallout 3, it will even more with this game. If it is very old, and you played a lot on it, it may actually be a hardware wear already. I remember I had to buy a new graphics card for my old computer after playing Oblivion for a year, because my old card started to overheat too much even with those games where it never overheated before, and I started to get freezes and blue screen because of it.
If your computer wasn't compatible with 64-bit systems, you wouldn't be able to install one. It is recommended to use a 64-bit system when you have 4 GB RAM or more. With 2 GB there is no point in installing 64-bit system, as some underdeveloped 32-bit applications tend to be more glitchy in 64-bit systems. But those are more rare with each year. Many viruses still can only infect 32-bit systems, like most porn-banners, which block your system trying to extort money for unblocking. But they are easily removed by any good computer technician most of the time.
By the way, I use 64-bit Windows 7 myself. FFXIV is the only game that makes my GTX570 to heat to 60 C degrees even when I set fans speed to 80%. With any other game it barely heats to 50 C with fans speed below 50%. For comparison, my old GTX260 heated to 60 C and more with most modern games even with fans speed set to 100%.