I’m so pleased that he’ll help you!
I had a styloidectomy with him, not a C1 shave - do you need both? Mr Axon puts a drain in to help with swelling and infection, that stays in overnight, that was probably the worst bit for me as it pulls a bit , is pretty uncomfortable and hard to sleep. Once that’s out the day after surgery then you should be allowed home. I took the painkillers they gave me in hospital, but didn’t really need anything stronger than paracetemol after the first couple of days. You’ll need to use an icepack around the area to help with swelling, and keep propped up too, but most of us with IJV compression are used to sleeping like that anyway! Eating/ chewing/ opening your mouth wide can be uncomfortable afterwards so it’s best to have smoothies or protein shakes or soft foods ready in case. (I had this for about a week after my first surgery, but not at all after my second!). And you’ll need to take it fairly easy for a couple of weeks, & don’t do anything strenuous or lifting etc for a few weeks, it’s hard to say for sure as everyone’s recovery varies.
There’s info in this section:
ES Information- Treatment: Surgery - Welcome / Newbies Guide to Eagle Syndrome - Living with Eagle
Hopefully others who’ve had a C1 shave will give you advice too, if you’re having that done as well. I think that there’ll be a bit more swelling with this, and you might find that the IJV compression symptoms ramp up a bit initially because of this, days 3-5 are the peak usually for swelling.
Let us know when you have a definite date! 
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