I think that the M575 are the of the newer type that require a bottom up bleed from the bled nipple (rather than a top down one from the resevoir like my old M585s used to). You really need a syringe. If you’ve got a basic Shimano bleed kit you’ve probably got a length of hose and a small dropper bottle of oil; I’d guess that this is the bottle you’re trying to attach to the bleed nipple. To do the bleed properly you’ll need a syringe filled with the oil in place of the bottle and you’ll also need one of these bleed tools for the lever end. If when you open the nipple there’s oil moving up the tube it sounds like you’ve got a filled system, it might be easier to drain it first and then bleed with new fluid.
Shimano have a handy video tutorial
Good luck, I dreaded my first hydraulic brake bleed but was forced into it when I inadvetently part popped a piston out whilst trying to clean it up. Once I’d managed it once it was fine (and I’ve since replaced calipers). If you struggle though I’m sure that the LBS would help you out.