How old? The Spanish first settled in the area of New Mexico about 1580; by 1610, Santa Fe was the capital of a province. Settlers from Mexico and Spain continued to arrive until the 1920s.
How old? 'Tis a puzzlement.

The song is known from Spain, 19th c, but Sephardic survivals suggest it is much older. Sephardic Jews were kicked out of Spain in 1492, although some took the cross, but were kicked out as well in the 16th c. Sephardic collections are from N. Af., and Mediterranean areas, where they re-settled. But let's settle for about 1870 for the early date, since that is the earliest students are sure of "Don Gato" in Spain. Are Sephardic collections pre-diaspora, or did they just pick up a good song at a later date?

The Ladino version given by Joe (no location) is very similar. This surprises me, since Don Gato is well-known in Mexico, and I would expect that it came from there. Dr. Carlos is in the Sephardic version I posted from Tetuan, North Africa.
North Africa, New Mexico, and ? (Joe's).

Yep, 'tis a puzzlement (guess what old movie I was looking at yesterday).