將這句講成這樣可以嗎？

No. With does not follow need.

The sentence should be interpreted as:
I need something which I can open this bottle with.
or
I need something with which I can open this bottle.

If you have a bottle opener, you can say something like this:

I opened the bottle with a bottle opener.

See, with is part of the adverbial phrase, with a bottle opener. With does not follow the verb need or open.