Watch it for a few days. If the needle goes up and down in line with your local pressure readings (which you can get from the internet) then I'd say no servicing is necessary. If the needle does not move then either the aneroid chamber is perforated or the linkage has seized.
For calibration, there will be a screw somewhere accessible, probably on the rear. If the linkage has seized, the needle won't move when you adjust that screw.
I would add to what Nick and Simon have said that an aneroid barometer should make you acquire the habit of being an "instrument tapper." The motions and forces inside the readout of an aneroid barometer are so tiny that you need to tap the instrument a few times before you read it. Usually, you will find that as you tap, the needle settles down and stops moving around. You may occasionally be surprised to find that the needle jumps quite a bit the first time you tap it. (Being an instrument tapper also is a good idea for many mechanical gauge readouts, like thermometers and hygrometers.)
Also a good idea not to be too dependent on your barometer for accurate comparisons with another source. Barometric pressure varies from place to place. The place where your smart phone gets its pressure reading might be quite different from your "local" reading. I have a local airport with a weather bureau station quite near me, and check readings with them via their website. Rarely do they match my barometer(s) exactly, but I don't assume that my barometers are malfunctioning because of that.
Sorry to see that the thermometer is missing. Note also that you have an altitude track. Surveyors used to use this type of instrument. It would be interesting to see the back and also to learn if the front bezel moves and adjusts the mechanism. In practice, you would adjust the pressure against a reliable standard, like a mercury barometer, then carry it to a new location. The altitude scale could then be used to tell you the difference in altitude at the two locations. But very frequent calibration is important for altitude measurements.
I was shocked at how easy it is to test them. The guy that has a shop full of them said to put it in a ziplock bag with as much air as you can, and then apply pressure to the bag and it should cause the needle to move. I tried it on 3 of mine and they all worked. The bad thing is where i live, theres never much change in pressure. Too boring to take up my wall space. lol
Your smartphone contains a sensor to provide barometric readings. It is sensitive enough to show the difference in barometric pressure between the top and bottom of a door frame. The resolution is far higher than any analogue barometer, either mercury or aneroid.
I always suggest it to teachers at school when teaching pressure. I use an app called phyphox to access the sensors on the phone.