Mongkok on Kowloon side tends to run fairly late.
You can always take a taxi up to the Peak for sunrise (but sunset is nicer). An early morning ferry from Central Pier might be another good way to catch the dawn.
Get an Octopus card and just hop on a night time bus and see wherever it takes you.
Early morning walkers -- starting well before dawn -- head up the Hong Kong hills, and you could join them. Maybe from Wanchai up to Magazine Gap, then around the trail to the left to Black's Link.
Note that moving back and forth across the Harbour can be laborious after the MTR closes, unless you know just where the overnight cross-harbour services depart.