Translation:Adamo always pours a lot of sugar into his tea.
Sia is the reflexive pronoun used in sentences like this one.
Say Adamo had a friend whose name was Jack:
Adamo ŝutas multe da sukero en lian teon. = Adamo pours a lot of sugar into his (Jack's) tea.
Adamo ŝutas multe da sukero en sian teon. = Adamo pours a lot of sugar into his (own) tea.
Well "a lot", "much", "little" are adverbs in English too. By the way there is no such thing as an "adverbial" case as opposed to the "nominative" case. Adverbs are invariable, therefore by definition lack any case ending. When adjectives are used to complete or describe a verb instead of a noun they become adverbs. So "The quick man acts quickly" or "I speak a lot". "Much sugar" is indeed an adjective but just like English prefers the adverb + preposition construction a lot + of so Esperanto allows for the adverb + preposition multe + da.
If made into an adjective then multa wouldn't need the da: it would be adamo sxutas multan sukeron.
Tea isn't the direct object. The accusative is being used to indicate motion: into instead of in. The object is multe da sukero, and since objects of prepositions that don't indicate motion cannot be followed by the accusative ending (in this case da), sukero does not become sukeron.
Can ŝuti be translated as "put" in sentences like this? I find that the only time I use the word 'pour' with non-liquids is when the act of pouring is happening as I'm speaking (so present progressive/continuous). If I'm not speaking in the continuous, then I would be much more inclined to say "Adam always puts a lot of sugar in/into his tea."
Edit: 5 months later and I have no idea why I asked this question. I no longer require an answer since I would actually use both pour and put interchangeably in this instance but can understand if they don't want to accept put as an answer.
The grammar of this sentence might make more sence if you translate it as: Adam muchly pours some sugar toward the inside of his own tea. The "toward" is the reason why "sian" & "teon" get the N's at the end; HOWEVER it still does not explain why "sukero" does NOT get an N at the end to indicate that it is the direct object. Please help me understand what's up with the N-less sukero!