Fancy your chances tests you on ellipsis, or omission of words.
As with many other languages, speakers of English quite often leave out words ~ most usually off the front or back of their sentences ~ where the circumstances are clear without those words, and/or perhaps urgent.
'Fancy your chances?' = '[Do you] think you are likely to do well?'
'On the blink' is a slightly old-fashioned expression, though still heard at least occasionally, meaning that a ~ usually, electric ~ machine is either not functioning at all, or only intermittently (i.e. any lights on it may only be blinking on and off, rather than shining constantly).