It is quite simple really. If you are LCY based and spend more than the stipulated number of days in the UK per year, then you are required to pay UK tax. The exact details can be found on the Inland Revenue website but from memory we are roughly talking about having to spend at least half of the year in the UK to qualify for paying tax. In the past, traveling days to and from the UK did not count towards this total but I am led to believe that days spent traveling to and from the UK do count to the total number of days spent in the UK. I suspect that there could well be a tax agreement between Ireland and the UK.
The days of using tickets or indeed rosters to prove when you were or were not in the country are numbered by the sounds of things. Someone could buy a ticket but never use it and rosters can often change, so the Inland Revenue are keen to see hard evidence such as copies of bank statements which can show if you withdrew currency whilst abroad etc.
If you elect to adopt the approach of 'they will never find me' and do not wish to pay UK tax when you are required to then I would advise you not to do this. CityJet is a well known brand and a large company so it is only a matter of time until the Inland Revenue are onto them so that they can check that all is above board with its LCY based employees. It is the crew members responsibility to sort this out so that they are paying the right tax.