For a five-minute walk (or even a longer ten-minute or fifteen-minute walk), pulling a small cart is not exhausting at all. I do it every week when buying food: I choose one of the several supermarkets in one of the nearby blocks, walk to it pulling my empty cart, after paying for the goods I put everything into the cart, and walk back home pulling the full cart. No public transport needed, though I've seen people carrying these carts into public transport too (this is easier when it's a low-floor bus, instead of the high-floor ones).
You can also get things delivered when it's a larger amount than can fit on your cart: while paying at the supermarket you ask for delivery, and they'll use a cargo tricycle to bring it to your building.