From the early 19th century, the phrase 'no bottle' meant 'no good/useless' According to The Oxford English Dictionary that is probably the usage that gave rise to sayings such as 'lose your bottle' and, more recently, 'bottled it'. In other words, if you lost your bottle, you had become useless.
It may also be derived from the Cockney rhyming slang words 'bottle and glass' meaning 'bottom'...ie the 4-letter word that opens with 'a', ends with 'e' and has 'rs' between them! One old meaning of 'bottom' was 'staying-power/courage', so losing it meant your bravery had gone.
In either case, you have "backed out of a situation", as you say, Sair.