In, on と at は感覚で覚えるしかないと思いますが、わかる範囲で回答してみます。

1.There are eight planets in the solar system.
何かの空間の中だからinかな。in the parkとかと同じ、多分です。

2.There are five players on a basketball team.
in the basketball teamでも有りと思います。
「in a team」はイギリスでは言われるらしいです。
北米では「on a team」のほうが普通と思います。

3. There are twenty-five letters in the English alphabet.
アルファベットはA～Zあって、その範囲内だからinかな？？これも多分です。

4. There are thirty days in September.
in は広い期間を表現するのに使います。e.g. in 2010, in the spring, in September.
短い期間、例えばFridayだったら、on Friday になります。

5. There are fifty states in the United States.
これもアメリカという範囲だからinでしょうかね。

因みにこのhttp://www.english-test.net/forum/ftopic50384.htmlサイトに下記のルールがありました（全部英語だし、他もいっぱいあるでしょうが、ご参考までに）：

In: –preposition
1. (used to indicate inclusion within space, a place, or limits): walking in the park.
2. (used to indicate inclusion within something abstract or immaterial): in politics; in the autumn.
3. (used to indicate inclusion within or occurrence during a period or limit of time): in ancient times; a task done in ten minutes.
4. (used to indicate limitation or qualification, as of situation, condition, relation, manner, action, etc.): to speak in a whisper; to be similar in appearance.
5. (used to indicate means): sketched in ink; spoken in French.
6. (used to indicate motion or direction from outside to a point within) into: Let’s go in the house.
7. (used to indicate transition from one state to another): to break in half.
8. (used to indicate object or purpose): speaking in honor of the event.

On: –preposition
1. so as to be or remain supported by or suspended from: Put your package down on the table; Hang your coat on the hook.
2. so as to be attached to or unified with: Hang the picture on the wall. Paste the label on the package.
3. so as to be a covering or wrapping for: Put the blanket on the baby. Put aluminum foil on the lamb chops before freezing them.
4. in connection, association, or cooperation with; as a part or element of: to serve on a jury.
5. so as to be a supporting part, base, backing, etc., of: a painting on canvas; mounted on cardboard; legs on a chair.
6. (used to indicate place, location, situation, etc.): a scar on the face; the book on the table; a house on 19th Street.
7. (used to indicate immediate proximity): a house on the lake; to border on absurdity.
8. in the direction of: on the left; to sail on a southerly course.
9. (used to indicate a means of conveyance or a means of supporting or supplying movement): on the wing; This car runs on electricity. Can you walk on your hands? I’ll be there on the noon plane.
10. by the agency or means of: drunk on wine; talking on the phone; I saw it on television.
11. in addition to: millions on millions of stars.
12. with respect or regard to (used to indicate the object of an action directed against or toward): Let’s play a joke on him. Write a critical essay on Shakespeare.
13. in a state or condition of; in the process of: on strike; The house is on fire!
14. subject to: a doctor on call.
15. engaged in or involved with: He’s on the second chapter now.
16. (used to indicate a source or a person or thing that serves as a source or agent): a duty on imported goods; She depends on her friends for encouragement.
17. (used to indicate a basis or ground): on my word of honor; The movie is based on the book.
18. (used to indicate risk or liability): on pain of death.
19. (used to indicate progress toward or completion of an objective): We completed the project on budget.
20. assigned to or occupied with; operating: Who’s on the switchboard this afternoon?
21. Informal. so as to disturb or affect adversely: My hair dryer broke on me.
22. Informal. paid for by, esp. as a treat or gift: Dinner is on me.
23. taking or using as a prescribed measure, cure, or the like: The doctor had her on a low-salt diet.
24. regularly taking or addicted to: He was on drugs for two years.
25. Informal. with; carried by: I have no money on me.
26. (used to indicate time or occasion): on Sunday; We demand cash on delivery.
27. (used to indicate the object or end of motion): to march on the capital.
28. (used to indicate the object or end of action, thought, desire, etc.): to gaze on a scene.
29. (used to indicate subject, reference, or respect): views on public matters.
30. (used to indicate an encounter): The pickpocket crept up on a victim.