Assuming "European" medieval studies, having some language skills (depending on area e.g. Old/Middle English, medieval/Insular Latin, or medieval varieties of the relevant European language otherwise) probably useful although these days I gather increasingly not something they can require due to them being less commonly offered. Doing your dissertation on a medieval topic and scoring very well in it would be useful I expect, especially if you do stuff involving methodological approaches in your area on primary sources (i.e. manuscripts/codices and the relevant text critical/codicological considerations) I expect is also good.
That said I imagine there are plenty of applicants with limited exposure to medieval history and medievalist methodologies and relevant language training who can also be successful so, not an absolute requirement. Obviously if you have those opportunities available to you, jump on them!
As for good unis for that area it may depend a bit on your particular area of specialist interest and region/period/topic. Generally speaking for Insular medieval studies in the UK I think ASNAC at Cambridge is pretty reputable. Oxford has an interdisciplinary masters in the field and I think has strengths in different areas there. St Andrews, York, and Nottingham also I think have a presence in medieval studies.