If the PSU is original, a new one would probably be necessary but to test the PC try borrowing one if possible... any PSU with SATA power connectors should work.
There is no Pentium D 965 3.4Ghz CPU... If it's a 965 it's probably a Pentium Extreme Edition 965 @ 3.73GHz.... and if it is a 3.4Ghz, it may be the Pentium D 950 @ 3.40GHz which is a little slower... the 965 is about as fast as a Pentium Core2 Duo 1.8Ghz, and the Pentium D 950 is comparable to the Core2 Duo T5270 @ 1.40GHz... they are not fast by today's standards but the PC may be upgradeable to a faster Cor2 Duo depending on the motherboard but with 6 RAM slots it should support the faster Core2 Duos.
As it is, it should be good enough for web browsing and general tasks...