I work in a charity shop. Customer demanded to buy a furniture item that had had the price label changed from 19.99 to 1.99. We think he had changed price but had no definite proof. It was explained to him that tag was wrong and item was actually 19.99, but he insisted the cheaper tag was already fixed on and he had seen item at that price earlier in day (we are sure it wasn't), and that was the price the item was and demanded to buy it.

Did he have right to do this or should item have been withdrawn from sale.

Neither! Say 'Someone seems to have changed the price' and firmly put a new tag on it or change it back to 19.99. I hope you didn't allow this fraudster to intimidate you into selling it for 1.99.
You aren't a big retailer with impossible to move stickers on your items. That's a provable mistake that a customer can hold the store to.
Hanging tags can be changed, and regular stickers can be moved from item to item. Let him try to sue you. He won't bother and he will lose anyway.

You just do not sell it for 1.99, esp since it appears to be a hand written price. It is very possible they changed it, and do it often.

Just be nice, refuse to sell it for the 1.99, and tell him it is for sale for 19.99.

First, any question on law needs to include your general locale as laws vary by area. While many areas have laws that require a retailer to honor the ticketed price, there are details specific to location. For example, I think a non profit might be exempt.