This is your first starting sentence:
It's time I went to bed.
The person who needs to go to bed is represented by a subject pronoun. All you need to do to apply it to a different person is to change the pronoun:
It's time he went to bed.
looking at the second starting sentence:
It's time to go to bed
Think of that as a complete sentence like
It is easy
If you want to add another noun (or a pronoun) to add further information to the sentence, you need to use a preposition.
It is easy for him.
Note that the pronoun following the preposition is an object pronoun. So, if you want to specify who needs to go to bed, you have to use a preposition with an object pronoun or a noun:
It's time for him to go to bed.
It's time for us to go to bed.
It's time for Jake and Mandy to go to bed.
Incidentally, "It's time to go to bed." does not mean "*It's time we go to bed." (which furthermore is not grammatically correct): it means "It's time [for unspecified people] to go to bed". If you said "It's time to go to bed" to a child, the child would take this to mean that only he or she should go to bed, and would assume that you had no intention of going to bed at the same time.