Step 1: Check whether the Assertion is correct.
Alcohols really do behave both ways, with a strong base they hand over the $O-H$ hydrogen as a proton, acting as an acid, and with a strong acid they use the lone pairs on oxygen to grab a proton and form an oxonium ion, acting as a base. So the Assertion holds.
Step 2: Check whether the Reason is correct.
The polar $O-H$ bond is exactly what lets the hydrogen leave as $H^+$, giving acidic behaviour, and the two lone pairs sitting on the oxygen atom are exactly what let it accept an incoming proton, giving basic behaviour. Both parts of the Reason are factually true.
Step 3: Check whether the Reason actually explains the Assertion.
Since the polar bond directly causes the acidic side and the lone pairs directly cause the basic side, the Reason is precisely the mechanism behind the dual behaviour stated in the Assertion.
Step 4: Choose the matching option.
Both statements are true and the Reason correctly explains the Assertion. \[ \boxed{\text{Option (A)}} \]