In our company we take turns to be on support, we generally have 3 x support phones, one on each network, and we get notified via sms depending on what is down. We do check each system where applicable by using the phones ussd short cuts, for some of our applications and in general, it is just to monitor the system. Each system will restart itself in the event of a connectio issue and sometimes we need to restart a specific tasks/job via dial up connection using VPN.
The support rota works nice as you end up learning more about everyone else’s jobs as well, and you do not have to be on duty 24/7, only once every other week. And in general you could plan ahead when going away or so.