Keep your fruit out of direct sunlight. Too much sunlight can speed the release of ethylene. Put your fruit in a large uncovered bowl in the darkest corner of your counter.
Put your fruit inside a perforated bag. The bag will ensure that the fruit does is not exposed to as much oxygen, while the perforations will allow moisture to be released rather than reabsorbed.
Keep the temperature and humidity as low as possible in your kitchen. You can't replicate the conditions of your refrigerator, but a cool environment with drier air will help preserve your fruit a little longer.
Store apples separately from other fruits. Apples release ethylene more quickly and can cause other fruits to ripen sooner than they normally would, if kept in close proximity.