Is It easy to buy back a put-option on its expiry date?

If the put is about to expire "with no value for the other person holding a long position" then it expires for you as well.

Worthless OTM options can usually be purchased for 5 or 10 cents very close to expiration. The incentive for the seller is to salvage some money. If you are rolling the short option out, you can often pay less than 5 cents if it's part of a combo order.

A more difficult problem is if it's late in the year and you want to sell a worthless option (to close) that has a zero bid so that you can deduct the loss this year. This can be done in combination with another option but it will cost 5-10 cents or so.

The option expires and disappears on the expiration date. That's it. Very simple.

By "sold a put" I think you mean "wrote a put" - which means someone else can sell their shares to you at a fixed price until the expiration date.

If that price (the strike price) is above the market price, they will make money by doing so and the options are "in the money." Your counter-party should exercise the options before the expiration to capture this money. Usually it's best to wait until the last moment to do so, but not necessarily.

Otherwise, if the strike price is below the market price, they will not make money by exercising and should just let the options expire. You still keep the option premium, of course.