"I will answer that question tomorrow."
You can have あした at the beginning, but then you need to use に (not は) with しつもん: あした、そのしつもんにこたえます。
The phrase is しつもんに答える, I am unable to explain why it’s not a direct object (を). Cf. in French, it’s also indirect object: répondre à une question (“answer TO a question”). You can also regard しつもん as the topic (は) and say しつもんはあしたこたえます.
"kotaeru" is what's called an intransitive verb, which means it doesn't take direct objects, unlike a transitive verb. Another verb that one might expect to be transitive and it's not is "au" (to meet). You typically (maybe always) use the "ni" particle with what otherwise would be the direct object. There are exceptions though. For example, you can use the "wo" particle with "tobu" (to fly), which is also transitive, but not for the direct object: "sora wo tobu" (to fly through the sky).
Maybe it's just a quirk of the verb こたえる then. In English also "we respond TO a question" (indirect object) but we "answer a question" (direct object, no "TO" required). Perhaps こたえる just doesn't take direct objects the way "to respond" doesn't in English. Thanks for the reply!
答える is an intransitive verb (自動詞) you can have 誰か
が答える (somebody replies); you cannot have × [誰かが]何か
を答える, that is ungrammatical.
But the particle に can be used for the target/purpose of the verb 質問
に答える (to reply to a question)
As for 明日 not being at the beginning, I think it is because of the choice of the topic:
- 明日はその質問に答えます : the topic is "tomorrow"; we are talking about "tomorrow", and, well, tomorrow (probably among other things) I will reply to that question.
- その質問は明日[それに]答えます : the topic is "that question"; we are talking about that question, and, well, I will reply [to it] tomorrow.