We eventually did our border hop to Arica, Chile to update my visa. The plan to go by car never happened, so to explain for the benefit of others without transport, here's how we proceeded. There are 2 Cruz del Sur departures from Arequipa bus terminal, at 7.00am and 7.30am. We took the second bus as the first was full, having booked our tickets about 3 days beforehand. The trip itself is straightforward, with one brief stop at a SUNAT checkpoint where you disembark to pass your bags through a scanner, airport-style. The ride to Tacna takes about 5 hours.
From the bus terminal at Tacna we took a taxi (4 soles) into the centre, next to the Plaza de Armas. There are plenty of hostels in the surrounding streets and we didn't pre-book. The 2-star hostels are basic, and the curtain rail fell off the ceiling in our room when I closed the curtains, but there was hot water. We'd hoped to get the afternoon train to Arica, but this being El día 28 de julio (Independence Day) it was closed. Instead we took another taxi back to the bus station and got on a bus to Arica for 10 soles each. The buses leave as soon as they are full, a system which runs pretty much like a conveyor belt.
Reaching the border we disembarked and passed single file through the Customs office, where I announced my overstay to the official stamping passports. This prompted him to leave his chair and walk over the road to get the "special rubber stamp" which he was not in possession of - the one that says "Salida por Exceso" - and proves that you have paid for your overstay. We pass outside, waiting no more than 10 minutes for everyone to board the bus. The bus conductors are very good, and nobody is left behind.
Arriving at Arica bus terminal is pretty uneventful. It's quite a distance from the town centre, unlike Tacna. We simply took a stroll down to the coast, about 10 minutes walking, and then came back and got on a bus. We were back in Tacna again after sunset, dropped our things at the hostel and then went out to eat.
We spent the next day sightseeing around the Plaza de Armas before returning to the bus station for the ride back to Arequipa. The bus departs around 2.00pm as I recall, arriving back in Arequipa at around 7.00pm.