"Except for" vs. "except on"

From each of your proposed sentences, I would assume that you have lots of time on Sunday, and very little time on any other day. If you want to convey that you have no time at all on any other day, you should say,

"Sunday is the only day when I will have time available to do the homework."

Actually, none of your example sentences is idiomatic American English. Ironically, however, the first sentence of your question contains your answer: "I have time to do the homework ONLY on Sunday and on no other day". You might want to shorten the sentence to:

I have time to do the homework only on Sunday.

"For doing the homework" sounds a bit strange to me.

I've hardly ever heard a native English speaker refer to homework as "the homework", unless you're talking about a specific piece of homework.

3 would be acceptable, and 4 sounds the most "native" to me.

I would suggest you rewrite the sentence entirely. As it is, you're using negatives to express what you want which is a bit confusing. I believe a positive would be more effective:

"I have time to do the homework only on Sunday."

3.Besides Sunday, I do not have much time available for doing the homework.

4.Other than Sunday, I do not have much time available for doing the homework.

Sunday is the only day available for me to do my homework.