In medieval times

This is a 1000+ year period so, unless you'd care to specify a more definite period (early, high, late?) and a loose geographical region it's difficult to give a response that is not so vague as to be of little use. When questions are posed like this there can be no definitive answer, and they tend to attract historical anecdotes or limited case studies which cannot be representative of the whole. I could load my response with chosen case-studies to make late medieval Anglo-French monarchs roaring tigers or meek pussy-cats. This would say nothing about the other c.750 years of history or various other geographical scopes.

The demands on a Frankish king of the 600s might be very different to the French monarch in the 1400s. Or the expectations and experiences of a Welsh ruler in the 1100s to an Iberian during the same period.

Here's my general and vague answer.

Martial ability in a monarch or ruler was generally desirable, but a ruler needn't possess those attributes to be lauded by his people or descendants. A proclivity for warfare might be encouraged but then so might a love of learning. It would depend on the individual and his guardians. Some heirs and rulers had the greatest warriors of their time employed to train them, for others we have no record at all, or the the individual who trained them was of little note.

Specific experiences, such as having to ransom your king after he was captured in battle, or his death in battle, or a major defeat when your king wasn't on the field might drastically change the expectations on his descendants. Other factors, such as fostering chivalric orders, might create expectations which later heirs had no ability to fulfil.

Ps. GoT draws very loosely from history ranging across hundreds of years, despite the opening premise echoing some aspects of the War of the Roses (pretty much only that there are families fighting - not exactly exclusive to the WotR). This is why I don't accept GoT as an indicator of geographical or chronological scope.