The web failover could be done through DNS (DNS Made Easy works at low costs). A file on your server with the string "UP" in it can be dedicated for this purpose, but a simple 200 (OK) result is also an option.
After a file for whatever reason becomes unavailable, the DNS no longer gives out the first IP address to your server, but switches to a second or third (IPv4 only for now), etc.
The mail failover can also be done with DNS Made Easy as they provide (2) backup Mail services to make sure e-mail gets delivered without any hiccups and gets delivered to your server when it comes back online (make sure your email server has all accounts set up before accepting connections).
For other services there must also be a creative solution using two servers and failover DNS or SRV-records (https://www.ietf.org/rfc/rfc2782.txt).