I would agree with ** that you should seriously consider whether it is worth relocating your family if you are really only going to be here one year.
If you still want to go ahead, yes, you need to start considering schools now. The best way to ensure a decent school is to find out whether your employer is willing to provide you with a corporate debenture, in which case you will almost be guaranteed a school place.
However, most people are not so lucky and will have to apply through other channels and most schools are over-subscribed so you are quite right to have concerns. You should start contacting schools now to find out what their waiting list situation, so you can make the decision whether to put your children on the waiting list. This usually involves a substantial deposit, so you are unlikely to want to do this for more than a handful of schools. You may not get places confirmed until well into the summer (this is because parents don't always give much notice that they are withdrawing their children) and you need to be prepared for schools having a place for one child and not for the other.
With regards to living arrangements, I assume you also know that HK apartments are likely to be much smaller than you are used to in NJ. If you want to stay on HK Island, then expect to get a two-bedroom 600 sq ft apartment for your budget. Commuting from anywhere on the Island to both Central (for IFC) and Quarry Bay (both of which I do frequently) is very quick and easy. North Point and Taikoo Shing are both in walking distance of QB and are popular with families.
You might want to consider the school place situation before deciding where to live, as given your children are quite young, if you end up putting your children in a school off the Island, for example, the American School in Tai Po, you may prefer to live nearby, which will be a longer commute for you (but HK isn't all that big and public transport is good), but you will get more housing for your budget and it will be easier for your children.
Good luck with your move.