Translation:This time I need to go to bed at nine.
I think there are probably some inaccuracies in the following, but it does set out the basic outline of the differences:
Надо and нужно are adverbs which are used only with verbs, and are mostly interchangeable, with some minor differences - надо is more direct, more "must" (without the option of not doing the thing that "needs" to be done), while нужно implies "really should" - I need to stop drinking coffee by 8 pm or I can't get to sleep. In this exercise, either надо or нужно can be used with the verb-phrase лечь спать "to lie down to sleep" = "to go to bed".
нужен/нужна/нужно/нужны is a short-form adjective used with nominative case nouns only. It often, perhaps always, is used as a predicated adjective, which is separated from the noun it modifies by a verb (which is made difficult to determine because the present tense of "to be" is omitted in Russian."
молоко нужно = "the milk is needed"
When you use an attributive adjective that attaches directly to a noun (the necessary milk), then there is a long-form adjective нужный whose ending -ый is changed to match the case/number/gender of the noun it modifies.
нужное молоко = "the needed milk"
The thing that confused me at first was the fact that Dative indirect objects are used with both the adverbs and the adjectives:
Мне нужно/надо спать = "sleeping is needed by me" "I need to sleep"
Мне нужно молоко = "Milk is needed by me" "I need milk"
мне нужна собака = "A dog is needed by me" "I need a dog"
Мне нужно/надо выгуливать собаку = "I need to walk the dog"