Three extra days before walking the Camino.

Are you flying in from the US or Canada? Many people are sleep-deprived and jetlagged after an overnight flight. It's not a bad plan to get started on your journey north on your arrival day since it probably won't be a great day for sightseeing, anyway. You may be able to nap on the 4-hour bus ride.

I see that a few of the ALSA buses stop at the airport, including one at 12:45 PM on the March weekday I checked. If that doesn't work for you, you'll need to get into town to the bus station on Avenida de America. It looks as if that will mean a Metro ride from Atocha Station and then a multi-block walk, all of which will take a good bit of time.

I imagine you're choosing the bus for financial reasons. A train would get you to Zaragoza in 1/3 the time, but the fare would be considerably higher if you bought the ticket after arrival in Madrid.

I hope you'll have the energy to see a bit of Zaragoza on your arrival day, but I know that I would not. Unfortunately, the bus/train station in Zaragoza is about a 30-minute walk from the historic center. The Aljaferia Palace is roughly midway between the two, but you may need to rush to get there before it closes. There is bus service from the station to the old-town area. Another thing to be aware of is that there aren't many food options near the station, just the basic sort of thing. There's a business-class hotel right in the station; I don't know whether there are less expensive options nearby. There's also tourist office in the station, on the upper level.

I haven't been to the French Basque Country and cannot comment on the tourist value of Bayonne vs. Zaragoza.

I think transportation between St. Jean Pied de Port and Bayonne is by bus rather than by train, though perhaps the bus is run by SNCF or a subsidiary thereof.

Sounds like fun and a good way to get over your jet-lag. Lots of people like to spend the day exploring St Jean before they start walking the next day. You will begin making friends everywhere you go. Are you on any of the Camino forums? Websites or FB pages? Do you have reservations at Orisson for your first night walking?

I love Zaragoza but going up to Bayonne will get you closer to SJPP. Bayonne has no real attractions and is known for chocolate. SJPP is very small and won't take long to see, but you'll have plenty of time to socialize and rest before the Camino. Buena suerte peregrino.