Having followed a professional golfer (family member) on many occasions, i can tell you that in theory, anything goes. I have seen 6 inch stilettos and bodycon mini dresses, all the way to cut off jeans and tank tops - and that is in the players lounge. However, sensible smart casual is your best bet. I have usually followed in hot steamy Washington DC, North Carolina-type weather, but as you are talking UK Ryder cup, I would probably go for nice pair of casual trousers, nice shirt and jacket or cardigan ensemble, with comfortable slip on shoes and a back up of rain jacket, gloves, umbrella and hat, with rain boots. If you want to look like you know what you are doing, you can wear actual golf shoes - generally, i think that looks affected and weird, but other may say it is sensible. Do not forget sunscreen for your face - with the possible sunshine and wind combination, it can mean sunburn very quickly!
Also, just to be aware, many courses do not allow mobile phones or cameras on actual competition days, although they will allow them to the practice and pro-am days.