Let’s say a player rolls and he lands on Park Avenue but doesn’t buy it. On his next turn before he rolls Can he buy that property? Or does he lose this opportunity?
No, that player may not buy that property. That property should have immediately gone to auction when the player opted not to buy it on their turn. Thus when they start their next turn, the property is already owned.
From the Monopoly Rules:
BUYING PROPERTY: Whenever you land on an unowned property you may buy that property from the Bank at its printed price. You receive the Title Deed card showing ownership; place it faceup in front of you. If you do not wish to buy the property, the Banker sells it at auction to the highest bidder. The buyer pays the Bank the amount of the bid in cash and receives the Title Deed card for that property. Any player, including the one who declined the option to buy it at the printed price, may bid. Bidding may start at any price.