I've used small 100 mm tall empty" Schwartz spice jars " for my oil reservoirs , put them in a 2" dia sleeve of rain ware pipe . After deciding the length of the plastic tube I pop riveted a bottom on the tube using a bit of 5 mm thick " L " shape aluminium that also had a 5/16 hole drilled in it for mounting purposes . After putting the tube on the left of the lathe by the chuck using one of the gear box bolts , I slipped the spice jar in & filled it with cutting oil ..it's out of the way & easy to use .
My application brushes are out of a set of 8 pigs hair brushes from the pound shop , I simply cut them down till about 40 mm sticks up above the rim of the spice jar .
Other oil jars / pots are slightly different in that the plastic tube has a hanger bracket fixed to them at the top with two 3 mm " Nyloc " nuts & screws for I always try to keep the bed , saddle & slide of the lathe free of anything .
The best thing that I read about and use on my mill and lathe now is any metal pot/tin used for oil, just drop a couple of neodinium (spelling might be wrong !! ) magnets in the pot. When you place the pot on any metal surface the grab is unbelievable. You will never spill any oil again. I used 2x 10mmx3mm magnets.