Not as long as he keeps this up.

こんにちは。

I can do nothing with him and he can do nothing with me, he thought. Not as long as he keeps this up.

と言っているところですね。
この Not をわかりやすくするために、この前の文をヘミングウェイさんに内緒で書き直してみます。

I canNOT do ANYthing with him and he canNOT do ANYthing with me, he thought. (We canNOT do anything with each other) as long as he keeps this up.
nothing = not anything（not any = no）っていうことなので、こういう言い換えが可能になります。

こうすると、この not が、前の文章を受けて、as long as ~のことじゃなくて、can do nothing、ってことを繰り返してるってことがわかりやすくなると思います。

つまり、As long as he keeps this up, we canNOT do ANYthing with each other.（解釈によっては、As long as he keeps this up, I canNOT do ANYthing with him. もしくは As long as he keeps this up, he canNOT do ANYthing with me. ）ということですね。