Kinda sucks you can't get a route map doesn't it! I searched long and hard for info on a route map on google and here on flyertalk but not luck so I'm pretty sure it doesn't exist.

I did find a thread here though with some off the top of head mentions of routes which I think was pretty comprehensive:

HKG-SIN
HKG-NRT
HKG-SGN

NRT-BKK
NRT-ICN
NRT-SIN
NRT-TPE

TPE-NGO

This being the UA own metal routes.