1) The transport problem from his house to nursery was your ex's problem to resolve, not yours. His sister needed to drop out, it was therefore down to him to organise alternative cover.
2) Nevertheless out of the kindness of your heart and mindful of DS's right to see his father, you offered to put yourself out by having DS and ex at your house the previous evening so DS could still see his father. (For what it's worth I would rather stick pins in my eyes than have my ex in the house for the evening)
3) He agreed to your kind offer.
4) He subsequently changed his mind and despite the hassle for you, out of the kindness of your heart and mindful of DS's right etc etc etc, you offered an alternative night.
5) He refused.
6) Therefore the onus on resolving the issue falls back to him. As he has done nothing towards that other than demand that you solve his problem for him by doing something that is both very inconvenient for you and not in DS's best interest, tough shit.
To sum up, YADNBU. He, on the other hand, is being a twat.
It might be worth suggesting to your mutual friend that, actually, you couldn't give a stuff what twattishness your ex is spouting on Facebook and that in future you'd rather not be told.
Some people seem to think they're doing you a favour by keeping you up to date with the latest pile of lies that ones ex is coming out with but I find life a lot less stressful and annoying if I can ignore all the drama.
Call his bluff - tell him that actually you think court would be a good idea, that a formal standing arrangement would benefit DS more than all this chopping and changing, and that you have tried to be as flexible as possible as you would like DS to have plenty of contact/father figure etc, but that he has to recognise that you also have a job and other obligations. He needs to only accept work that he can manage alongside his obligations to his son. Frankly [unfortunately for your DS] he sounds like the type that will back off as soon as it looks like any sort of effort. I would put all of this in writing and keep a copy in case it ever does become legal so that you can demonstrate your reasonableness!