This is the plot of Gilgamesh the Immortal, where Gilgamesh is granted immortality by Utnapishtim the immortal (in the real Sumerian Gilgamesh saga, Gilgamesh did strive for immortality, but failed to achieve it).
Usually just how immortality is achieved is left unsaid; the most popular choices outside magic and magic-like "science" (i.e. a handwavium immortality serum) are self-repairing nano-machines.
This is because "old age" is a composite of cellular disrepair, DNA degradation, and toxic substance accumulation in the cells and in the organism at large. Nano-scavengers would be able to undo all these damages.
Then there is the - possibly minor - problem of memory. Human memory is holographic in nature, likely composed by networks of competing potentials responsible for both storage and recall. Accumulating lots of new memories would tend to wash out older, less-networked and less-recalled memories. On the other hand, maybe an alien "doctor" made up of nanobots might be able to reengineer the storage and recall "algorithm" making it more efficient, or even interface with it and supplement the human memory with an additional artificial storage system. Even if this cannot be done, the most recent century's worth, perhaps more, of memories would be useable without troubles; forgetting the older information might actually be an advantage.
Other ways of achieving the same goal are usually semi-handwaved "methods" that "reengineer" the organism, rendering it both resistant to damage (including long-term damage from environmental poisons and UV rays) and immune to sickness and cancer. This is sometimes done by tricking the cells into believing they're still in the neotenic phase - the organism is kept in the young "developmental" stage, which slows more and more, never reaching adulthood and triggering the aging process (this approach does no longer match current medical research though - it is now clear that "aging" is not a process that starts at a certain point and can be delayed or prevented; it has the advantage that immortals are never fully sexually mature, so cannot reproduce and overpopulate). This is exactly what happens in Damon Knight's 1957 The Dying Man.
Another way (just for completeness) is divine will. This is what happened to Casca Rufio Longinus, the soldier who speared Jesus Christ on the Cross in Casca the Immortal.