Question

The rabbit...(run) as soon as it saw the old man

Answer 1

$\huge \boxed{ \mathcal{Correct \: answer }}$

$The \: rabbit \: \underline{ ran} \: as \: soon \: as \: it \\saw \: the \: old \: man$

Answer 2

The rabbit started running as soon as it saw the old man.

This is a past continuous tense that gives the sentence meaning and a format because the rabbit started running and then was continuing the act as per the statement of the narrator.

So the exact meaning is conveyed when the sentence is narrated using a past continuous tense in the context.