Last year I bought a small olive tree. I kept it under the sun the whole summer taking it inside the room during colder nights. It was more bushier then but with some time the leaves started falling down. I stopped watering it too much and it stopped dropping leaves. When colds came I took it inside and left it standing in the most sunniest spot in the apartment. It started to emit some sort of sugary white residue on the leaves and some leaves became dry but still green. I am at a loss what's going on because I am watering the tree once a week but maybe not sufficiently? Any advice would be welcome.
Your plant has a scale infestation - I can see a couple of small dark brown disc like objects low down on a woody stem, and the white fluffy deposits in other p laces are the young, immature scale.
Spray with neem oil, including backs of leaves - you can pick off the hard,dark brown discs on the woody parts, or touch each one with a cotton bud dipped in alcohol to make it easier to remove them. You will need to keep an eye on it and treat with the spray probably 2 or 3 times to keep it in check. General info on olive trees and pests here https://www.olivegroveoundle.co.uk/olive-tree-pests-diseases/