Bangkok is a sprawling city with tons of opportunity to get your cultural fill. Around every corner is someplace to explore, so the question simply becomes, how much do you want to spend doing it? If the answer is nothing, then that is perfectly reasonable.
One of the great free things to do in Bangkok, regardless of budget, is visit a few of its majestic temples; luckily, entering most of them comes at no expense. Head over to central Bangkok near the famous Paragon shopping center and see Wat Patum Wanaran. Other free temples of interest include either Wat Mangkorn Kamalawat in Chinatown or Dusit’s Wat Indraviharn.
I did mention a shopping center here, and one of the great free things about Bangkok is spending a few hours running between them. Nestled in one spot of town is Central World and Paragon. They have tester treats at all hours of the day for you to enjoy. If you tried hard enough, you could enjoy an entire lunch worth of diverse foods in the food court, from sausage to soups.
With dozens of cultural and art centers about town, many of them come at no expense and they are certainly worth at least the price of admission. For some great modern art, the Queen Gallery presents a rotating platform of regional masterpieces as well as from local artists. Other free museums include the Silpa Bhirasri Memorial & Museum as well as the Bangkok Art & Culture Center.