Yes, you could walk from the Airport to the Camino. Since you are flying in, why not bus or taxi to Irun for the night?
I would stay one night in Irun. Then get your "bragging rights." Train station in Hendaye is unique. People get off the train, or sometimes rush off the train, from France and do a direct walk through the small station to the Spanish side. The rail lines are parallel to each other. So the off on again is a direct shot. But you have to cut through the small station building.
Why? cause this was the border control point years ago. Especially under Franco in the Basque country. And, "the station is a border railway station where all trains have to stop, as those coming from/going into
Spain have to change gauge from 1,435 mm (4 ft 8 1⁄2 in)
standard gaugeto 1,668 mm (5 ft 5 21⁄32 in)
Iberian gauge. The electric pickup supply also changes here from 1500V DC (overhead France) to 3000V DC (overhead Spain). Between the stations of Hendaye and
Irun, both track gauges run together."
Then start the Camino in the morning.
Stop by the chapel on the way to Pasajes. Can't miss it.
Then walk from Irun to Pasajes (Pasaia) The Alburgue there is nice. It is at the upper elevation of the town. So to go eat is a nice walk down the steps and back up again. No stores, bars, cafes, etc where the Alburgue is located at.
You will have to catch the small ferry (Pasaia) to cross over to the other side to continue the walk in the morning. Once off the ferry on the other side, then decide which way to continue on your walk. Off the ferry boat to the left is walking in the city. This is side walks, road, stores, apartment buildings, etc all the way to Sebastian.
Or, off the boat and to your right is the hills and woods way.
Irun:
The airport is walking distance from center of Irun. Yes, A walk....but can be done.
In Irun I stayed at the Pension Bowling (no bowling alley there LOL). Not a bad place. There were other walkers staying there. The TI office in Irun is great. She will give you maps, directions, help etc.
The first few days will be "tough" ... well for me being over 70 the elevation gains and drops were a killer.
I just cannot get enough of San Sebastian (Donostia). So I stayed there two nights.
I made it to Gijon. Due to leg injury I called it quits there. And rode bus to Ferrol (where I grew up).