To control the beginning and end of a note, the technique of tonguing is used. To prepare for this technique whisper the sound "taa". The sound begins with your tongue sitting behind your top teeth, blocking the passage of air, and you make the "taa" sound by quickly withdrawing it, and letting a stream of air begin from your outgoing breath.
The next step is to do this with the tin whistle in position to play a B note, with your tongue lightly on the hole in the mouthpiece. As you withdraw your tongue, the note will have a well articulated beginning. The "t" part of the sound gives the note a definite starting point, and the "aa" part of the sound keeps your throat open so that the flow of air remains constant and the note sounds even.
To end the note, you put your tongue back on the mouthpiece rather than stopping your breath. This will end the note as crisply as it started. It is worth practicing the tonguing technique many times on a single note until you are comfortable with it.