Cost really depends on how much detail and how long it takes, unfortunately cheap rarely means good. Expect to pay upwards of £50 an hour at least, your probably looking at upwards of £600 for the half sleeve, depending on how much detail/how fast your tattooist is/how big your arms are. Try and find portfolio's of all the artists in Manchester and Liverpool and take a while to see not only who does the most impressive work but who can do what you want well, for example there's a great japanese work guy in Colchester, but he's shit at portraits. Also Circle Takes The Square rock the house.