We occasionally do the old HDD to new PC trick - always works as long as you're not going from IDE to SATA or vice-versa as the PC will not boot (or if you're on WIN2k). If it's IDE to IDE it will boot 99% of the time and give you the opportunity of installing the chipset, lan, audio, graphics etc drivers which I would suggest you download from the PC manufacturer beforehand.
The long/short beep you describe might be a cpu or case fan issue (not connected, not working properly or missing from under the hard-disk as in the case of some Dell PCs).
If you prefer to start from scratch get yourself an IDE/SATA to USB converter so you can offload the data you need - see
Plexus USB to IDE & SATA Adapter With Power Adapter (Supports 2.5" / 3.5" IDE & Serial ATA Hard Disk) - Ebuyer for an example.