"Do you think you will be able to get a flight for early next week?"
"Do you think you will be able to book a flight for the start of next week?"
"Do you think you will be able to get a flight for early next week?"
"do you think you will be able to get ~" 「〜取れそうですか?」
"get a flight" 「飛行機の予約は取る」
"for early next week" 「週明けの」
"Do you think you will be able to book a flight for the start of next week?"
"to book a flight" 「飛行機を予約する」
"for the start of next week" 「週明けの」
[これも](https://eikaiwa.dmm.com/uknow/questions/60040/)役に立つかもしれませんので是非クリックしてください。
Do you think you'll be able to reserve a plane ticket for the beginning of next week?
Do you think it's possible to book a flight to New York early next week?
この場合、次のような言い方ができますよ。
ーDo you think you'll be able to reserve a plane ticket for the beginning of next week?
「週明けの飛行機のチケットを予約できそうですか?」
Do you think you'll be able to reserve ...? で「…を予約できそうですか?」
the beginning of next week で「週明け」
ーDo you think it's possible to book a flight to New York early next week?
「週明けにニューヨークへのフライトを予約できそうですか?」
Do you think it's possible to book ...? で「…を予約できそうですか?」
to book a flight to ... で「…行きのフライトを予約する」
early next week「週明け」
ご参考まで!