I completed around 940km (almost to temple 66) of the Shikoku Henro in Oct/Nov 2013. Because of time constraints, we walked 2/3 of that distance and took public transit for the rest, spending 31 days total. It's an amazing trip and I really want to go back and complete it/do it again.
I went with my then-girlfriend and we both wore Merrel Ascend Gloves, which are a non-GTX, lightweight mesh zero drop trail runner with a rock plate and light padding. We were both avid hikers/runners conditioned to using these types of shoes- I wouldn't recommend a zero drop shoe to someone not used to them, but I would recommend a lightweight mesh running shoe to anyone. Having a shoe that drained and dried really quickly was key. GTX would have been miserably humid, gotten wet anyway and never dried. Our shoes retained very little moisture, allowing us to regulate the temperature and wetness of our feet with socks only. Always keeping a pair of dry socks to change into was important. Wool socks only! These will keep you warm while you walk even if soaked unless conditions are really extreme. You will get used to having damp but warm feet- this is totally OK as long as your shoes can drain/dry out and you can put on dry socks when you sleep.
Also remember that this is definitely not a wilderness experience- you will almost always be near a shelter/temple/convenience store/cafe/restaurant where you can take a break and warm up if needed.
We did find out that foot fatigue and soreness limited our daily distance. In our case this was due to all the road walking and having 35ish pound packs (this was before I started reading BPL)! If I were to do this trip again, I would pick very similar shoes maybe with a little more cushioning but definitely with a much lighter pack (thanks BPL!). Maybe even a more road-biased shoe as the dirt trails are by no means treacherous. By and large the dirt trails are very well kept (and have been that way for hundreds of years). Some of the mountainous areas (thinking of the approaches to Iwayaji, Yokomineji on Ishizuichi-san, and Unpenji) can get muddy and slippery depending on the weather, but it is overall such a small proportion of the pilgrimage that I would not get really aggressive trail runners just for those areas. The tread will be worn down by all the road walking before you get to the mountains anyway! Since you are not conditioned to walking long distances, I would emphasize finding a shoe that is really comfortable and has enough cushioning to prevent foot pain from walking on concrete/asphalt. Since this stuff is really personal, the best you could do is buy one or two pairs of shoes that seem good and do as much training as you can with them. 10 mile road walks with a pack would be perfect- do a few of those and you will know a lot more about your feet and shoes which can then inform what you actually take to Shikoku. Like any trip, if you can train and gain experience beforehand you will be much more likely to be successful.
1 pair of shoes lasted the trip for us and would have gone the whole distance, but all shoes are different so it depends. There is so much road walking that tread wear is not a huge deal, as long as the shoe doesn't actually fall apart. I wouldn't bring another pair of shoes anyway, saving weight is especially important to keep your feet/joints happy walking on hard surfaces. Don't count on being able to find the same exact shoes in a store in one of the cities, although there are stores in the cities where you could find trail and running shoes. Keep in mind most non-Japanese outdoor gear is much more expensive in Japan (at least compared to the US). MontBell stuff is reasonable and readily available, though. Not sure about having something shipped to you while on the trail, we didn't try that although I'm sure it is possible if you know how to navigate the Japanese postal system.
Can you tell I really like talking about this trip??
I'd be happy to answer other questions or give general advice about Shikoku, in this thread or by PM. There are many many ways people do this pilgrimage so a lot of more specific recommendations depend on how you plan on sheltering, whether you will be walking only or taking other forms of transportation, your language skills, and general comfort levels about hiking/walking and travel in other cultures.