The correct option is
A Preposition of Place
Prepositions of place are used to show the position or location of one thing with another. It answers the question 'where?' Some examples are, across, as at, in, on, while, during, near, over, under, between, behind, etc. According to this explanation, option C is the correct answer.
Prepositions of time are used to indicate the time of an action or time relationship between nouns in the sentence, such as at, to, in, etc. Since the sentence is not talking about the time at which the event happened, option A is an incorrect answer.
We use preposition of agency to show that something causes another thing in the sentence. Examples are, by, with, etc. As the sentence does not mention the reason why the cat ran across the road. option B.
The preposition of manner is used show how something is done. Some examples are. by, like, as, and so on. According to this, the word 'across' in the sentence is definitely not a preposition of manner. Hence option D is incorrect as well.