"He patiently waited for her arrival."

The correct answer is Option (4) → Adverb of Manner

An adverb is a word that tells how, when, where or how much something happens. It can also describe an adjective or another adverb, for example she sings sweetly, very hot.

The adverb here is patiently. It means calmly, without getting upset. It tells how he waited.

Option	What it tells	Example
(1) Adverb of Reason	why something happens	The road was blocked and therefore we took a longer way.
(2) Adverb of Time	when something happens	I finished my project yesterday.
(3) Adverb of Place	where something happens	Put the plants outside in the sun.
(4) Adverb of Manner (answer)	answers how?: it shows the way an action is done	Dadi listened patiently to all our complaints.