What is the contingency/continuity plan if the single controller becomes incapacitated while on duty with no warning to pilots?

Same as if the radios stopped working or otherwise communication fails. Execute the planned procedures (which vary).

Often Approach will take over the "tower" and operate in crippled mode (no clearances to cross active runways, you must go down to the end kind of thing).

Some airports are uncontrolled at various times and would revert to that. Some airlines would require the pilot execute a missed approach and deviate to a towered airport, others may allow them to land.