Usually you would have a second DNS resolver configured in /etc/resolv.conf (or whatever name resolution config system you are using, resolvconf, systemd-networkd, etc). The system will fall back to this resolver if the first resolver fails to respond (and/or replies NXDOMAIN, I'm not sure. The exact order and fallback conditions may vary depending on which system you use). This can be another dnsmasq instance, a public DNS resolver, your ISP's resolver, etc. This allows at least basic DNS resolution to work before your dnsmasq instance comes back up.
I would also add automatic monitoring for dnsmasq (either check that the service/container is running, or check the TCP connection to port 53, or check that DNS resolution is working for a known domain, etc)