how to upgrade Aperture 3 purchased in US but now living in a different country

Aperture is no longer developed and sold by Apple, so it cannot be purchased in any country.

Which version of Aperture do you currently have installed? And which version of OS X?

If you are running Yosemite OS X 10.10.3 or any newer system version, you need the update to Aperture 3.6. This version is not sold outside the Mac App Store. If you purchased your original Aperture 3 version from the Mac AppStore and Aperture is showing in your purchase history, you can try to update by reinstalling. But updating the boxed DVD versions of Aperture is only possible, if Apple Support will help you.

See: Updating to Aperture 3.6 After the Release of Photos

Aperture is no longer available in every country, it's finished.

Which version is it, and which version of OS X do you have

You need version 3.6, which is no longer available (I believe) but maybe someone more knowledgeable with Aperture will come back.

You're welcome.

All Aperture updates released since Mavericks have only been available from the App Store and not at the software downloads page, Csound1. There has been a very short window of opportunity, right after the Mavericks release, when it has been possible to associate the Aperture and iPhoto versions purchased on installer media with a registration key with the AppleID and to convert them to MAS versions. This window of opportunity has been closed, when Apple pulled Aperture from the AppStore.

A few OPs managed to talk Apple Support into providing a redeem code, so they could download Aperture 3.6 from the AppStore, but there is no guarantee that all Apple support will help in this case.

Any owner of Aperture 3 should be entitled to a free update to Aperture 3.6, it is an expensive professional application, but it has been more than a year that Apple stopped the Aperture development.

Updates are no longer possible at all, only reinstalling and thus replacing the current version by Aperture 3.6.

See my modification of Barney's iPhoto User Tip for Aperture:

Updating to Aperture 3.6 After the Release of Photos

Maybe this can help you:

Download Aperture 3.4.5

Maybe this can help you:

Download Aperture 3.4.5

Aperture 3.4.5 runs well on MacOS X 10.8.5 Mt. Lion, but to run Aperture 3.4.5 on Yosemite you have to jump through the hoops. It does not launch on Yosemite and we can only run it by opening the package contents of the application. It cannot be updated from the App Store to the current version.