There's been plenty of Giant Monster vs Giant Robot fights, it's almost its own genre. What started the trend? What was the first Giant Monster vs Giant Robot fight?
Godzilla vs Megallon is a film from 1973 that features a human size robot called Jet Jaguar. This robot is able to grow to a gigantic size if needed.
Prior to this there's also a 1967 japanese movie that features King Kong and his robotic nemesis Mechanic Kong: King Kong escapes.
But the earliest depiction of a giant robot fighting a giant monster I can find in japanese productions is the 1956 comic (and 1964 anime) Tetsujin 28-go, known in the USA as Gigantor (by the way, probably the first giant robot of japanese pop culture). He fights a gigantic octopus in his episode "Gigantor Vs. The monster of the deep".
Probably the earliest widely seen show with a giant robot fighting giant monsters was Ultraman.
From the link:
Ultraman (ウルトラマン Urutoraman?) is a Japanese Tokusatsu television series that first aired in 1966. Ultraman is a follow-up to the television series Ultra Q, though not technically a sequel or spin-off. The show was produced by the Tsuburaya Productions, and was broadcast on Tokyo Broadcasting System (TBS) from July 17, 1966 to April 9, 1967, with a total of 39 episodes (40, counting the pre-premiere special that aired on July 10, 1966).
One day, Shin Hayata, the Patrol's most capable member, was patrolling near a lake when he witnessed a space monster named Bemular, who had come to Earth in an orb of blue energy, being chased by an orb of red energy. Hayata also gave chase to Bemular, but the orb of red energy collided with Hayata's ship, causing both to crash land and killing Hayata in the process. Ultraman, who was generating the red orb, felt guilty for killing the capable scientist/soldier, and to make up for the terrible thing he had done, chose to revive Hayata in exchange for him becoming his host, who would contain him while he was on Earth.
From that day forward, Ultraman staunchly defended Earth against aliens and monsters, with Hayata and the SSSP assisting him.
I remember watching these as a kid and loving them.