Step 1: Read the sentence.
The sentence is 'He shot a bird ____ an airgun.' We must fill in the right preposition.
Step 2: Find what the airgun is.
The airgun is the tool used to do the action of shooting. So we need the preposition that shows a tool or instrument.
Step 3: Recall the rule for tools.
When an action is done using a tool, instrument, or weapon, English uses 'with'. For example, $\text{She wrote with a pen.}$ and $\text{He cut the paper with scissors.}$
Step 4: Test the other options.
'By' is normally used for a doer or a means of travel, not a hand-held tool. 'For' shows purpose, and 'in' shows place. None of these fit a tool.
Step 5: Place the right word.
Using 'with' gives 'He shot a bird with an airgun', which clearly shows the airgun as the tool.
Step 6: State the conclusion.
The correct preposition for a tool is 'with'.
\[ \boxed{\text{with}} \]