Can I purchase the Windows 8 upgrade and install it on another PC?

Yes you can! (Providing you have Windows 7, excluding Windows 7 Starter Edition).

According to the faq for the Windows Upgrade Offer:

Do I have to install the Windows 8 upgrade on the eligible PC, or can I install it on another PC?
The upgrade offer is tied to a Windows 7 PC purchased between 2 June 2012 and 31 January 2013, but you can choose to redeem the offer and install the upgrade on any compatible Windows-based PC with a qualifying base operating system.

Are all PCs purchased with Windows 7 between 2 June 2012 and 31 January 2013 eligible for the Upgrade Offer?
Eligible PCs are new PCs purchased during the promotion period with a valid Windows 7 OEM Certificate of Authenticity, product key for and preinstalled with:

Windows 7 Home Basic;
Windows 7 Home Premium;
Windows 7 Professional; or
Windows 7 Ultimate
Note: Windows 7 Starter Edition is not eligible for this offer.

While you can use the upgrade on another PC, also see What happens to the Windows 7 key when upgrading to Windows 8?.

After you upgrade, the license for Windows 7 is no longer valid, and you no longer have a license to use Windows 7 on the original PC.

Does your friend have a Windows 7 license he can use instead? (Perhaps from a PC which will use the upgrade, if it is not OEM).