How can I download previous iTunes purchases onto a new iPod using someone else's computer?

the safest way would be to create new user account on that computer and download and sync your stuff. Account can be deleted after that.

You never need to buy stuff you already paid for from Apple. Once you paid you are authorized to have it on up to 5 computers and 10 devices at the same time. Just open itunes, sign in to your account and use instructions here.

http://support.apple.com/kb/ht2519

As has been pointed out, if you live in a country that allows access to iTunes in the Cloud, you can redownload most of your pas purchases. However, if you do this on someone else's computer will you lock them out of thier Apple ID (iTunes Account) for 90 days. So I advise against it. I suggest you just wait until you have access to your own computer again to sync the iPod.

That is why another user account.

It doesn't matter: http://support.apple.com/kb/HT4627.

Once a device or computer is associated with your Apple ID, you cannot associate that device or computer with another Apple ID for 90 days.

A computer is associated to an Apple ID by MAC Address, not account. So if the OP associates their friend's computer to their Apple ID and downloads past purchases that computer (MAC Address) will then be locked to that Apple ID for 90 days and the friend will have to wait to associate it back to their Apple ID.

Not by Mac address, but by UDID, but does not matter. Remember the times when family had to use the same computer, but multiple apple id's. They could do that with multiple accounts and every account would stay with it's own apple id. Same situation, but I am not going to argue - have it your way.

Multiple people can indeed use multiple Apple IDs on the same computer in multiple user accounts as long as they do not use iTunes Match, automatic downloads or re-download any past purchases. However, if you'll read the KB article I posted a link to you'll see that for the purposes of those services a computer can only be associated with one Apple ID every 90 days. If the OP associates their friend's computer with their Apple ID and downloads purchases from the purchase history the owner of the computer will be locked out from using iTunes Match, automatic downloads or re-downloading any past purchases.

It's not that the friend will be unable to use iTunes at all, but what the OP wants to do will associate their friend's computer to thier Apple ID for 90 days which may cause the friend some inconvinence.

Did you even read original post? OP already gets the message that she can't change id. All she needs to do create another account and try. She will see the same message if you are correct and she will succeed if you are not. And once again you can have it your way.

http://support.apple.com/kb/ht1495

Glad you got it sorted out.

No worries, even if it does affect him/her, one call to Apple will get your friend a relieve of that 90 days rule.