Hi Sarah,
I first travelled with my now 10 month old son when he was 9 weeks and it was the easiest journey! The older he's got the more active and less likely to sleep for long periods! and it's even easier if you're breastfeeding as no sterilisation or baby food to worry about. so I would say travelling with a 4 month old wouldn't be that hard here. Even if you're running around a lot, the chances are he'll sleep - of course it all depends on your baby's personality but I found (and still find!) that the busier I am, the happier the baby is - he just loves to watch the world go by! and if he's on my front in a baby carrier he's happy for hours.
You would have to think of a few things though. For starters, will you have a car and driver here while you are looking around? If not, you should know that most taxis don't have seatbelts (admittedly there are lots of new ones for the expo but I haven't managed to catch one yet and it's never guaranteed that you'll be able to find one when you need it!). A lot of people don't bother with cars seats here but it all depends on how comfortable you are with that. An alternative, if you're carrying your baby is the metro (I wouldn't recommend it with a stroller) which is clean and easy but maybe a bit daunting for your first visit! A stroller is a must to bring (even better if you have one that has a car seat attachment) and I found it really helpful to get my son used to his travel cot before we left so that he was in familiar surroundings wherever we were. I would also highly recommend a baby carrier of some sort (I use an Ergo which I love! and I have a very big baby!) - there will be lots of places that you won't be able to take the stroller and it's much easier to just pop the baby in there and carry him.
A couple of things that might help you if you do decide to bring your baby and/or move here:
www.shanghaitoyclub.net is a great resource for hiring baby equipment (car seats, travel cots, baby carriers etc if you don't want to bring them all with you)
I'd also recommend that you join the ShanghaiMamas yahoo group and post your questions on there - it's fine to join if you're just thinking about moving here. They are actually starting a new website soon but it's worth joining the yahoo group until that happens.
http://groups.yahoo.com/group/ShanghaiMamas/ (click on "join this group" in top right corner and follow the instructions - don't let the long application put you off - it's totally worth it!)
Hope some of that helps in your decision - if you have any specific questions please feel free to pm me
Hangzhou