I had downloaded Ubuntu the 64-bit version of 12.04 but I have an old computer which does not meet the minimum requirements for 64bit. Can I Install the 32bit version from this download or will I need to download the 32bit version?
You will have to download the 32-bit version. The 64-bit version is functionally identical, but contains binary files which are entirely different.
Basically - no, you cannot install a 64bit O/S on a 32bit host.
The O/S image has been compiled specifically to the processor architecture.
In the murky world of virtualisation, it is possible to run a 64bit O/S virtual image on a 32bit host - this is explained in the linked question.
Thus - if you are using a 32bit O/S such as Windows 7 - then install virtualbox and then install the 64bit O/S.
Since you mentioned you have an old computer - this may not be appropriate since you may be short of resources in terms of CPU strength, RAM, disk-space etc.
You will need to download the 32-bit version and install that.
Newer processors are 64-bit capable and you can install either the 64-bit version or the 32-bit. The 64-bit version is recommended because it makes better use of the processor. It can handle wider numbers and has more registers so these can be used to produce faster code.
If you try to do it the other way around it just wont work as a 32-bit processor won't understand 64-bit code.
As a side issue since you have an "old computer" you may want to consider a lighter variant such as Xubuntu. This will run faster on older hardware and is also supported here.