In Ulysses's case, the principal problems of old age appear to be boredom and querulousness. At the beginning of the poem, he applies negative descriptions to everything around him in Ithaca. The island is covered with "barren crags" and his people are "a savage race." Presumably, these are circumstances which have not changed much since he was a young man, but they irritate him now.
It is clear that Ulysses misses adventure and, at the end of the poem, his solution to the problems of old age is to behave like a young man and go off in search of adventure. His quest this time is genuinely new. Before, he had a specific reason to travel to Troy, and refusing Agamemnon's request for help in the war against Troy might well have had negative consequences. This time, he does not know where he is going, and the only reason for his voyage is to seek adventure.
This quest for uncertainty and danger is Ulysses's solution to the problem of growing old, but it is also a more general solution to the lack of an aim or mission in life. According to Tennyson's classical sources, Ulysses was reluctant to go to Troy and frustrated by the length of time it took him to get home, but these experiences seem to have changed his character permanently, making adventure necessary to him.