MrHaha:
Good question. I always have another person at the other end to do the change at the remote site. However, if I have to do it with one person only. I would add the route first then change the IP address of the outside interface. I also would add your site's external IP as a allowable SSH IP address for SSH into the ASA.
For router, you could set the router to reboot at 15 minutes before you make any changes. Make sure you don't do a WR then the router would fall back to the old address. Then if your changes failed. At the least you could remote back into the router after 15 minutes because the router would reboot.
I don't know if the ASA has the same commands. Also, if your ISP is changing IP address. I think you need to coordinate your changes with the ISP at the same time unless you have two circuit to switch from old to new. If you have two circuits to switch, you need someone to be at the remote site and physically unplug the old circuit then plug in the new circuit. Thanks twice before you proceed!!
KC
KC