A slight correction:
to get 1m accuracy you will need something capable of sampling at 300MHz.
That would give a 1m resolution
, not accuracy (there's a huge difference).
To get anything usable for the purpose, the sampling speed should be in the GHz range.
you will need access to a proper surface mount workshop to deal with these sorts of timing accuracy.
And proper microwave CAD and techniques, to avoid standing waves, lumped component interference etc.
I have seen seasoned engineers pull their too early greyed hair, trying to make PCB's for practical µwave implementations.
I would definitely not let a beginner even ponder doing this - it's the fastest route to disappointment.
I'll still suggest Sharp distance sensors as they will work.
U.S. could be used as well, but will require synch'd pulses from all "towers" and an omnidirectional receiver (or a large array to simulate omnidirectionality) on the 'bot.