No, 百円 is not the abbreviation of 一百円. The basic rules students learn at school are:
- Always append 一【いち】 before 万, 億, 兆 and other larger four-digit grouping units
- Do not append 一【いち】 before 十, 百 and 千
See Wikipedia for examples. (Particularly, note that 1000 is always せん, not いっせん, according to elementary textbooks)
If you want to practice, use this paper for fourth graders.
If you are an advanced learner, you may have actually heard いちじゅう (10), いっぴゃくまん (1,000,000) and so on. But they are basically jargon used by bankers, brokers, military members and so on. They even say ふたじゅうふた instead of にじゅうに to avoid any confusion. It's like saying "alpha" instead of saying "A".
As an exception, for some reason, いっせん is already popular in daily conversations among ordinary adults. You can forget the textbook rule here and say いっせん whenever you feel like saying it explicitly. But never use いっぴゃく and いちじゅう; they are usable only in special contexts, but simply wrong otherwise.