There are many potential problems with what you are doing.
Yes, it is very likely an old WinME PC will not have the gigabyte of memory needed to properly run Vista. In fact, it probably has less than 128M.
Then, the hard drive might be too large for your PC to fully recognize. You will need to look into the BIOS and make sure it sees the full size of the hard drive.
Assuming you even had the minimum amount of RAM, Vista will not be expecting new hardware and will try to load drivers that might lock up the computer. The video and sound drivers are the most likely culprits, so boot into safe mode and disable those drivers and try booting normally again. If you can boot normally it will take several reboots as Vista finds the new hardware and loads drivers. You will then need to look for hardware that Vista could not find a driver for and hunt for those yourself.
Oh, and if you get past all that, you will need to authenticate Vista before it will fully run. If the hard drive has an OEM version of Vista you will not be able authenticate it since you are not licensed to move an OEM copy of Windows to new hardware.