I can't tell you anything about the cartridge, others here can.

I can tell you about the rifle itself. The rifle is not a Martini Henry. The Martini Henry is a Martini actioned rifle with a Henry Rifled barrel. Most often seen in the form of a British Military rifle. There are also commercial variations using the Martin action with Henry barrel.

You Rifle is a Greener-Martini Sporting rifle.

I have two with this style action configuration. They are my favorite pattern Martini's. One is Marked Greener and per Greener made in 1898, and chambered in .500 EXP. (577/500 No.2 Express.) The second is marked made for Rawbone and is marked for .303 ammunition. This one uses bullets that are .303 in diameter and not the .311 of the British .303 cartridge.

As I said others will have to comment on the .360 Express, but what little I do know it is in the range of what would make a good whitetail deer cartridge here in the US.