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When asked by the pharmacist why I needed a syringe, I replied: to apply grease to mechanical parts. I know these don't have a curved tip as mentioned by Hungry in another M14 lube thread, but it is all I could find.
If you're in one of the cities that has a Lee Valley or some other serious wood working stores they have the curved tip ones that they sell as glue applicators.
The other thing I've done with syringes for this sort of thing is to get some brass tubing from a hobby shop, cut off a couple of inches, anneal it with a torch then drill the end of the syringe tip and push in the tubing and curve it to my requirements. Again this was originally done for glue but it would work for grease as well. Some 3/32 OD tube is what you want.
If you are ever in a hobby shop that sells balsa models they'll have little plastic extension tubes that will slide over the end of the syringe. The tubes are needle thin and about 2 inches long, really handy for tough to reach areas and good for hitting the exact spot. Ask for applicator tips for CA glue, they should know what you want. They are a couple bucks a dozen.
not sure if the lube you are using is a petroleum product, but the syringe in your pic, is going to have the rubber stopper turn into goo after one use, if it is (I speak from experience, using Moly in the same model syringe)