Cold start issue

Hi, I have recently acquired a 1.6 petrol Octavia on a 55 plate (Classic) which seems to have a bit of a cold start problem. When starting from cold engine will crank and start for a second or so before cutting out. Always starts up the at the second attempt but will sometimes run a little rough at idle for a few seconds. If i feather the accelerator when I start her up then will usually start first time. Also if I drive to work (about 20 mins) and then turn the engine off and on again will start up first time no problem. If car sits for 30 mins or so then will usually have the start issue. There are no warning lights on. Does same with both keys and always starts second time so don't think it is a key issue. Battery is new and of correct spec etc and charging system seems to be working. I do not think the engine is getting enough fuel from a cold start but unsure how to diagnose this further. Anyone goy any ideas as to what may be wrong or anyone had a similar problem?