Hi there. Can you post a picture of your budgie from both sides and from the front? Is it possible that your budgie had a stroke? (it sounds like you are describing generalized weakness in one area)
It isn't uncommon for budgies to close one eye while resting if they feel uncomfortable in an environment - it helps them to still see, but they only have to exert one eye. If it's constant, that's a different story.
First, I would clean the eye with saline solution and a chamomile infusion (steep chamomile in boiling water, leave it to cool with a plate over the top of the glass, and then dip some gauze or kitchen paper in the mixture and hold it up to your budgie's eye). You can use the chamomile every 8 hours as it helps reduce inflammation.
If that's not the cause, I would say that a vet visit is necessary.
Also look at the other symptoms of your budgie, particularly for symptoms indicating respiratory infection (breathing heavily, tail bobbing, puffed up, sneezing a lot or producing wet sneezes). Sometimes budgies express respiratory infections through their eyes.
Best of luck and let us know how you get on!