Bus in to Burgos

I think your best bet is to get a taxi from Ages to Burgos. When I was there a few days ago there were plenty of walkers taking taxis into the city as it was pouring rain. I'm sure you'd have no problem getting someone to share with you.

There's a bus to Burgos from Agés and from Atapuerca but just on Fridays. Take a look at www.autobusesjimenez.com for info about schedules, fares and ticket purchase for the Fridays' bus from Agés and Atapuerca to Burgos.

If you go through Castañares, there are urban buses from Castañares to Burgos (line number 4 and line 16). And if you go through Villafría, there are urban buses from Villafría to Burgos too (line 8, it runs daily every 30 minutes on working days, every hour on Saturdays and every 2 hours on Sundays and public holidays). Urban bus so it can't be booked in advance. You pay the driver on the bus.

Orbaneja Riopico and Cardeñuela Riopico are served by on-demand bus service (aka transporte a la demanda) some days of the week. Those are buses thought to give access to locals of rural areas to some public services (e.g.: to go to the hospital), need to be booked (minimum) a day (24 hours) in advance by phone (the bus won't stop at a town if nobody booked it that day...) and I don't know what are the rules for tourists. The phone number used to book the bus in advance can be used too to get info about the bus but, as it's a service aimed for locals, the person on the phone may speak just Spanish. Out of Atapuerca and Agés, it might be some on-demand services too (in addition to the Autobuses Jiménez bus I quoted above). You may try your luck once on the ground asking locals if there's a bus on your travel date, what's the number to book it... but there's no guarantee you'll be lucky so have a plan B.

There are city buses from near the train station near the airport. That is not quite 15 k away, though. There are buses from Belorado to Burgos, but I don't know if they go through smaller Camino villages between Belorado and Burgos.

The walk into Burgos if you go the River Path is one of the nicer sections.

Here is Ayuntamiento de Burgos website regarding transportation including urban buses and the bus station.
http://www.aytoburgos.es/movilidad-y-transporte