This depends on what you mean by "prince" since this title refers both to a local hereditary ruler - not necessarily related to any kings, as well as a heir to the royal throne of a king, who did not have to rule anything.

The autonomous ruler-prince would divide his life between his family life and managing his private affairs, and local administration. The latter would involve everything from settling land disputes, managing taxation, to diplomatic correspondence (and were talking here about a lot of letters and wax tablets to recieve and send: enough that most princes would destroy several brass seal-stamps from overuse, and the work of the principal cancellarium required its own industrial supply of parchment).

Also, prior to 1300ties, many rulers would manage their land by traveling between households of their vassails (they were called princeps ambulans - the traveling prince for that). This means that travel would take a lot of their time.

To add to /u/Feevoulous's comment, the question posed is incredibly broad. What can be termed "medieval" spans 1000 years between the 5th and 15th centuries, I believe (European history is not my area, and my level of education in history is limited to an AA).

Also, the term "prince" can refer to a variety of individuals, including as Freevoulous mentioned, hereditary rules, heirs, or retroactively applied to members of non-European social structures that held similar powers or social influence to European princes.

Furthermore, details of daily life would vary depending on locale. An 11th century Prince of Antioch would have a routine vastly different from that of a 15th century Scottish prince, or an 8th century Moorish Prince.