Well, my son isn't a year old yet. A lot of what we during the day is determined by the weather. In the summer it is usually ghastly hot or nasty and rainy (at least it was this past summer).
If the weather permits, my son likes to go out in his stroller. In Hong Kong there is usually a lot to see--almost to the point of over-stimulation for a baby. Lots of lights, sounds, people, smells. When going out pack sunscreen, mosquito repellent, lots of water, snacks and everything you might need.
If you have room for an inflatable swimming pool (or you live in a nice place with a pool) swimming or sitting and play in, I know my son loves to splash around in the water--and with the heat in Hong Kong water is a good idea usually.
Hmmm....I know that a lot of people get their babies involved in classes--language classes, yoga classes, playgroups with a special focus. I think that these things can be a good idea for babies but I am torn as I think that some parents are just trying to make their kids into "mini geniuses" and there tends to be a lot of pressure on kids to be smart and perfect here in Hong Kong and I don't want my child exposed to that too early. Nonetheless I'm sure there are some wonderful schools and groups out there that provide an outlet for babies and mamas.
We just started a forum/group for young moms in their 20-30s who have young babies. You're welcome to join this group that plans to get together and have some activities. There are other playgroups centered around different interests and residential areas in Hong Kong that may interest you as well.
Also, Hong Kong has several beaches that are pretty accessible if your child likes to play in the sand and such. There is Hong Kong Disneyland and Ocean Park, two theme parks that may have things that are interesting for your child and you.
I personally don't get the chance to get out and do much with my son but he enjoys playing at home with his toys and going out for strolls. Those simple things keep him occupied.