I don't have a yahoo account, but it sure looks like you can sync your notes to yahoo.
Do you already have your yahoo account set up on your iPhone's Mail? If not, do the following:
Go to Settings > Mail, Contacts, Calendars > Add Account and choose the yahoo button and add the account.
Whether you have it already set up or not, once it's set up, check there (in Settings > Mail, Contacts, Calendars > [your Yahoo account] and see whether there's a way to select yahoo notes to be used.
Assuming there is a way, they'll be synched when you do your sync (not continuously as iCloud would be).
I have everything set up just as you described but the notes are not syncing either way. I have created new notes in both yahoo notepad and on my iPhone after resetting the account up and everything else is syncing except the notes. As a matter of fact now the contacts are duplicated as many times as I have "reset".