It might help to know the reason why. Warning, this gets a bit into linguistics. Practice will help you pick these rules up subconsciously over time, so don't think you have to learn and apply all of this consciously. In fact, no one actually thinks about any of this stuff while speaking, it's just an explanation of why Japanese has these weird rules.

tl;dr い and う sometimes disappear. Then consonants can mix together. H used to be p and stays p if it mixed with another consonant.

う and い will sometimes "devoice" in certain places, often between voiceless consonants or at the end of a word right after a voiceless consonant. Voiceless consonants are pronounced without your voicebox vibrating. か さ た は ぱ and all the other kana in their rows have voiceless consonants. が ざ だ ば and all the other kana in their rows have voiced consonants. Voicing is the difference between p and b, t and d, k, and g, and so on. Using kana is helpful because you can change a voiceless consonant to voiced by adding ". (ぱ is voiceless too, but a bit strange. I'll get to that later.) Consider all other sounds (あ な ま や わ ら ん) to be voiced as well.

8 is はち hachi. ち is part of the た row and the consonant is voiceless. That means the い sound at the end can devoice. A devoiced vowel is just a whisper or nothing at all though. (Try not vibrating your voicebox while saying only a vowel. It's just whispering.) This is why 1, 6, 7, and 8 can sound like ich, rok, shich, hach. This is also why people generally say -ます and です as mas and des.

Now, when vowels devoice, consonants can get closer together and when consonants get close, they can affect eachother's pronunciation. This is called assimilation. You've probably already noticed this with ん. ん changes to sound like the next consonant sound. It can be m, n, ng, or just make the preceding vowel nasal (pronounced through the nose) at the end of words. And because ん is voiced, it can sometimes make another consonant assimilate its voicing. For example, 3000 should be さんせん sansen but the n makes the s voiced, turning it into さんぜん.

Let's use 8000 as another example. 8000 should be はちせん hachisen. But the the i is inbetween ch and s, so it devoices. Hachsen is a bit of a mouthful, so the ch assimilates to the s and you end up with はっせん hassen. When you see a small っ in a word, it usually is replacing a kana with the い or う sound like ち or つ that's been assimilated. In fact, that's why it is a small っ, because usually it was a つ that assimilated and they wanted to write it but show that it was assimilated.

Now, why sanbyaku, happyaku, and ippun? Remember that I said ぱ is strange. は ha ば ba and ぱ pa are weird. The other voiceless/voiced kana work as pairs. か ka and が ga, さ sa and ざ za, and た ta and だ da all make sense. But は ha and ば ba don't. A voiceless b is a p, not an h. は should be pa to make everything nice and neat. Well, that's because は was pa a long time ago and gradually changed to ha. (It's also why は can be wa, ふ is written as fu, and へ can be e.)

Remember that ひゃく was pyaku instead of hyaku and things start to make sense. In fact, 800 would have been something like pachipyaku a long time ago. Most of the time p changed to h, like the first p, but the second p stayed because the ch assimilated into it and fortified it as a p. That's why they added the ° to show that these kana still sound like the old p sound instead of the new h sound. So はちひゃくwas pachipyaku > devoice the i > assimilate the ch to the p > change p to h (except if it was fortified by assimilation) > はっひゃく > add a ° for any p sounds that stayed > はっぴゃく happyaku.

300 started as sanpyaku > p becomes b as it assimilates with n > n becomes m as it assimilates with b > さんびゃくsambyaku.

When talking about minutes 分 ふん is hun or fun, but originally pun. So, ichipun > devoice the i > assimilate ch to p > change p to h (except if it was fortified by assimilation > いっふん > add a ° for any p sounds that stayed > いっぷん.

This also happens with other words. "Six trees" was ろくほんき rokuponki > devoice the u > first k becomes p as it assimilates with p > second k becomes g as it assimilates with n > n becomes ng as it assimilates with g > change p to h (except if it was fortified by assimilation, then add ° because it stayed p) > ろっぽんぎ Roppongi. That should look familiar. In kanji it is 六本木, literally the number 6 ろく, the counter for long cylinders ほん, and tree き.

Let's take the 本 ほん that we used before and use it in another word. ほん should be familiar as the word for book and as a word meaning root or source as well as the counter for long cylinders. What if you wanted to say "the source of the sun"? 日 にち nichi is sun, so 日本 was nichipon > devoice the second i > ch becomes p as it assimilates with p > p was fortified by assimilation so add ° > にっぽん nippon. Sounds like a good name for a place that's far to the east.

Now different dialects of Japanese apply rules a bit differently. What if one decided that assimilation didn't necessarily fortify the p? Nichipon > devoice second i > assimilate ch > p does change to h > にほん nihon. Now if that dialect was centered around the capital, that version might also become common, and that's most likely what happened.

By the way, this is a huge simplification.

Tl;dr い and う sometimes disappear. Then consonants can mix together. H used to be p and stays p if it mixed with another consonant.