Ooh exciting. I think try and treat the little pony as much like you would a 'proper' horse from day 1 of arrival (so insist on manners when in the stable, tied up, leading etc)and resist temptation to spoil with too many cuddles and treats etc (hard when they are so damn cute!). Probably teaching you to suck eggs here but this time of year can be dangerous on the fresh grass both from the point of view of laminitis prevention (always an issue with the diddy ones) and also avoiding too much freshness/energy, so if you have a choice put him out in a slightly barer paddock or maybe invest in a grazing muzzle?
As with any horse I give them a few days to settle and am certainly very tolerant of any minor mishaps in the early days but I do expect them to get on with their job fairly quickly, generally the sooner they're into a routine the quicker they settle. So on day 1 it would be no more than a quick lead up and down and then into a stable or out in field to settle, then day 2 maybe a groom followed by a walk in hand or a quick lunge, building up from there onwards (depending on his current fitness), and I'd aim to have the kids up well within a week assuming you have a well-fitting saddle for him already (just as important for a little one as any other horse).
Feel free to lunge before they ride to take the edge off, and keep ridden sessions very short and sweet to begin with, stay well within their/your comfort zone so if it's a pootle in walk for the 4yo for 10 mins only the first time that's great. Keep every experience a positive one for the kids and the pony and gradually extend things as he settles. At that age you really want riding to be fun and they tire/get bored easily so formal 'lessons' on position, reins, steering etc are minimal and I like to try and play little games, 'I spy', Heads, shoulders knees and toes, can you ride without reins, can you ride with your eyes closed, little handy pony type tasks like picking up something from the fence, carrying a cup of water without spilling it etc, - obviously discourage too much shouting and wiggling about while pony is still new so it needs to be quiet/calm - later on providing pony is bombproof they can do walk 'races', see who can do round the world quicker etc etc. I'd get them out and about asap to keep things fresh although you probably want to stick to riding in an arena/at home the first few times.
I don't know if you usually lead off the bit or a headcollar or the noseband (I don't like the latter at all) but I'd clip onto the bit at least at first to be sure I was in control at all times and def test his manners and do a little reminder session if needed before you put the kids up.
Don't be afraid to ask for help if needed, and are you planning on still getting lessons for them when the pony comes? I would find a good instructor that will come to you, even if it's more for your benefit than the kids atm! A professional eye and rapid sorting out of any problems is so worth it for little ponies when you can't jump up yourself to sort any issues, and I know from bitter experience it's really hard to teach your own kids more than the basics as they usually don't want to listen to Mum! Also, if you can identify a confident slightly older child that you can have on stand-by to ride occasionally that would be good as I find every pony, particularly one whose regular rider is a novice, benefits from being schooled a bit by an experienced rider...