君 - male suffix, casual, diminutive
山本君は - (so, who do you think Yamamoto is? A? B? C? See below for the answer.)
気になる - be on one's mind
存在 - existence
気になる存在 - irresistible, inevitable, you cannot just pass (this person) by
一度 - if there's a chance/opportunity
誘う - invite, court, woo, call on, ask someone out, lure, seduce
と - you use と here to mark what you are thinking, like a quotation mark or thought bubble
思っているところ - thinking right now.
- ところ doesn't mean place here, or not in a literal sense. With verbs, tokoro indicates the closeness of the action to the present time. It refers to the state of the action, whether it is just past, current, or just about to happen, and it emphasizes this. Here, used with the present progressive, it means he is in the middle of thinking (something). Depending on context and the tense of the verb, ところ also means having just done something or being just about to do something. More on this at JapanesePod101.
一見 - at first blush
遊び人 - a person who is not serious
割と - relatively
ガードが固い - to be well-defended, so as to let nothing in, to have one's guard up at all times
甲: Say, what do you think about Toshiko?
Yamamoto: I can't stop thinking we should date. Right now I'm thinking, just once, I want to ask her out.
甲: Ohhh. You know, at first she looks so carefree and easygoing, but actually, she's a tough nut to crack, relatively speaking.