Civ3 isn't going to care how much RAM you have as long as you have 1 GB. It simply isn't that RAM-intensive of a game. Civ4 would be another story, but with Civ3, you don't need to worry about how much memory you have. Pull up Task Manager at any point that it's running, and you will never see it using 1 GB. For example, the 180x180 late-industrial game I have up now is sitting pretty at 300 MB.
One issue might be that you simply have way too large of a map size. What size map is taking this long?
There are some techniques such as disabling air and/or water trade that can drastically speed up the game. Another technique is disabling maintenance on city buildings. There's at least one more that escapes me at the moment, but the folks in Creation and Customization would be able to suggest some more. It won't help in a game that's already in progress, but would in future games.
The processor does matter, a ton more than RAM does. How many cores it has doesn't matter - Civ3 will only ever use one core at a time - but how good the cores are does matter. You'll notice a good difference between, for example, a dual-core Intel Atom and a dual-core Core 2 Duo (the latter being much better). So it's possible that your dual-core processor isn't that well-suited to large Civ maps.
And as to Windows 7... yeah. I've stuck with XP in part because Civ3 worked better with Vista (and by association, Windows 7). Civ3 is usually pretty stable (unless there's a bug in a mod), so crashes do seem odd. If I had to run Civ3 on Windows 7 I'd be sure to not install it to C:\Program Files (or C:\Program Files (x86)), and instead to somewhere like C:\Civilization III. That way at least you avoid any of the protected folders "security" features that can interfere with Civ3. But beyond that, I abandoned Windows Vista and 7 far too long ago to help.
Do you have another (even if older) computer that you could test the same savegames on? That could also help tell if it's just way too overwhelming of a game for any computer, or if the computer is the problem.