The Babylonian Talmud, in Tractate Chullin (24a), notes this contradiction in starting age and reconciles it by saying that Levites entered training at 25 and began to serve at 30. (From this we learn that if we don't see signs of progress with a student in 5 years, we don't expect him to succeed.) However, this restriction applied only to the mishkan, the portable sanctuary in the wilderness, because then the job of being a Levite included constructing and assembling the structure. But in the temple, the Levites' duties aren't so physical; they involve singing and assisting the priests. The talmud learns that this applied only in the wilderness from Num 4:47 -- "to do the work of service and the work of bearing burdens".
Chullin also addresses the upper boundary, saying that Levites in the temple (but not in the wilderness) continued to serve so long as they kept their singing voices. The importance of the voice is derived from II Chron 5:13 -- "And it came to pass when the trumpeters and singers were as one to make one loud sound" -- this is talking about the Levitical choir.
But we still have a problem: there were way more Levites than Levitical jobs. And further, the Levites were given cities throughout the land of Israel, but commuting to Jerusalem each day wouldn't have been practical. If they all served in the temple and lived in Jerusalem then what about their cities, and if they lived in their cities then what about the temple service? The talmud (Ta'anis 4:2 (26a in the Bavli)) says that they and the priests were divided into rotations, so any given Levite didn't work all year in the temple. Therefore (to get back to your original question), it's not just a question of what he did in his youth and old age -- most of the year he wasn't working in the temple, so what was he doing?
This is speculation, but the fact that the Levites were given cities (instead of land for larger-scale agriculture) suggests that they had ordinary occupations in those cities, probably including teaching torah. Their cities included surrounding fields, so some (smaller-scale) agriculture was also possible. Levites too young or too old, or disqualified due to disabilities, or ones not currently serving would all go about their non-temple lives.
Finally (h/t Fred for pointing this out), the Rambam (Sh'mita v'Yovel 13:12, based on Deut. 33:10) says that the Levites were tasked with teaching HaShem's "upright ways and righteous laws to the public". This could obviously be done anywhere there are people, which certainly includes the Levitical cities.