Upvote:2
There might be no buses between Ancona and Fano, however there seem to be buses between Rome and Fano. According to Rome2Rio these are operated by Adriabus and depart daily from Roma Tiburtina train station. The total travel time is of about 04h30m which looks shorter than your train journey with the included 1h30m waiting time in Ancona.
Should you prefer taking the train to Ancona, your options for getting to Fano seem to be train, taxi, or rideshare (blablacar or equivalent).