Is buying an apartment at a certain price and selling it to a buyer at a higher price after taking possession of it considered usury, or is it a permissible murabaha sale? And does the ruling differ if the seller only buys half of the apartment?

There is no objection to the Murabaha sale to a party who commissions a purchase, provided that the Sharia-compliant conditions are met. It is permissible for you to ask him to buy the apartment or half of it, with your promise to purchase it from him at a profit, provided that the purchase takes place after the seller has acquired ownership of the item and it has come under his responsibility.