It means "I didn't feel like very much (of something)". The something is omitted for poetic reasons.
What might that be? At four o'clock you might be expected to be hungry, so you don't "feel like very much (to eat)". In this form it is used informally, e.g.:
"What do you want to eat?"
"I don't feel like very much" (to eat)
So he is copying that form, but changing it.
- I didn't feel like (I was) very much (of a person)
- I didn't feel like (doing) very much
- I didn't feel like (eating) very much
- I didn't feel (feelings) like very much (in particular)
(i.e. not happy, sad, excited etc.)
This is Leonard Cohen. What exactly it is he doesn't feel like is omitted to keep the listener suspended. The suspense is resolved in the next sentence.
Four o'clock in the afternoon
And I didn't feel like very much.
I said to myself, "Where are you golden boy,
Where is your famous golden touch?"
In this case I think he means he doesn't feel like doing or creating very much, even shaving. He is depressed.