Congratulations!
I live here, have a 19mo DD. As with everything, the answers to your questions depend on how much money you're prepared to throw at them. 4 yo DS will be the right age for pre-Kindergarten. You are probably not eligible for for public pre-K (as it's known), and even if you were there are way, way fewer spots than children. I'd forget it if I were you. You could stick with daycare or nursery. Don't know about the former but the latter could cost you anything from a few thousand $ for the year, to twenty-odd grand. In sought after areas, you are not guaranteed to get a spot even if you have the money, but everyone finds something. Frankly, given the huge upheavals your children will go through, I imagine they won't need the extra boost of the top level daycare/nursery. It will all be a learning experience, anywhere clean and convenient should do!
5mo DD could go to the same daycare/nursery as DS. Oddly, you stand better odds of getting one of them in because you'll be wanting to send them both in.
Reading the above through, it sounds a bit scary. I don't mean it to be! It can all be done, and you would have an amazing time, but it will pay to do your research from home, and do as much of the ringing around/emailing admissions people before you get here. I would be optimistic, you will definitely find something that fits.
You could google daycare/nurseries in midtown (assuming you will be living nearby, as many UN employees do), and take it from there. Bear in mind applications are going in now for the Sept 2014-2015 school year, so make clear this is an exceptional case. Also, I think the UN has its own resources. Other websites are UrbanBaby.com (but beware, this site makes MN look like Netmums, it's brutal!), MeetUp (for when you get here), schools.nyc.gov.
A year here doing your dream job would be AWESOME. This is a great city, and (contrary to popular belief), also great for young children. Anything is possible here, if you want to come you can, and have a great time.
Good luck!