There is no reason why it wouldnt work. The socket is the same, and core 2 requires less cooling, with the exception possibly being a quad. It's not going to hurt to try if you monitor your temps.
The cooler will fit yes. The question I would be asking myself is "will it cool enough?". Depending on your casing, you could consider getting a good 120mm tower cooler like the ones from Xigmatek. If you get one that does both LGA775 and LGA1366, the cooler could even be re-used on an i7 build. Such a cooler would also alow you to OC your processor and get more for your money (OCing a Core2 processor is ridiculously easy).
OCing = Over Clocking...
higher speed, higher clock and higher voltage from your current state(CPU in this case)
But if u do that then u just shortening your cpu lifespan...