Literally, arsenal means a collection of weapons, or a place where weapons are stored. This sentence is using a figure of speech, or a non-literal meaning, so it isn't actually referring to weapons. Here the word is comparing the speaker's knowledge to weapons. If his mind is his collection of knowledge, it is his figurative "arsenal". So he didn't have a single answer (or weapon) in his mind (arsenal)

Inside/in/into are all fairly interchangeable, and refer to the interior. He went into the house/ he went inside the house, etc On/onto usually refer to being on top of something, or being on the surface. He walked onto the roof, she stood on the bridge. Cross is a rarely used abbreviation for across. Across refers to distances. He swam across the pool. They walked across the pier. Round is a rarely used abbreviation for around. Around is a vague word that can imply a circular motion or a general direction. They walked around the puddle/ She walked around the park. The different connotations of prepositions are confusing even to English speakers, so I wouldn't worry about it too much. Just know the general idea of the word and look at example sentences. Some English prepositions make no sense at all