Is there a way to get rid of a lump on my eyelid without surgery?

Maybe: It all depends on what the lump is, but if it is solid chances are it would need to be surgically removed. Of course this can be done with relative ease and minimal discomfort, depending on the size of the lump and where it is on the lid...

Depends: It depends on what the lump is. Some lumps can be treated with medication and some require surgery. I would recommend seeing a board certified plastic surgeon or dermatologist to determine what it is and the best course of treatment.

Depends: If the lump represents a benign growth it may need to be excised (removed surgically).

Maybe: It is decided by the cause of the bump. If on the surface and not dangerous, some will treat skin "bumps" with liquid nitrogen or topical treatments. If due to laxity of skin/septum and represent fat pad bulges, the treatment is reduction of the fat and allowing the septum to tighten and hold the pads inside the orbit.

Warm wet: Compresses and tinter of time.

Depends: If it is a sty, there is some chance to reduce it with hot compresses. If it is a skin bump, it can be benign or cancerous. So that you don't take a chance with a cancerous growth, you should show it to your ophthalmologist. In these case it's better to remove it surgically and as earlier as possible.

Depends on cause : Yes, it depends on cause. If it is due to blepharitis, simple hygiene may suffice. However if it is due to a tumor that should be biopsied, deferring surgery may not be in your best interests and could actually do you more harm in the long term.