Can you improve a cheap cooler's performance with reflective tape and foam board?

Yes, line the inside walls and lid with Reflectix or rigid foam board, taped with aluminum foil tape, and add adhesive foam weatherstripping around the lid rim to create a real seal. Pre-chill the cooler and use block ice instead of cubes. This costs about $15 and two hours, getting close to Yeti performance for a weekend trip.