I don't know about the probability of flight cancellations, but your idea of spending the afternoon sightseeing in Lima is solid, especially if you are energetic. Downtown Lima is probably the most important place to visit, and it will be roughly 45 minutes from your air bnb. Uber cost approx 30 soles. See downtown, leaving at around 5pm to beat the traffic (and touristic locations shut down around that time anyway) and head to the district of Miraflores for a nice meal, drinks, or even dancing or live music, depending on the day of the week and your energy level. Just be sure to stay hydrated for you trip the next day, and don't over eat, since the altitude can kick your butt.
If you can afford to splash out, try Restaurant Central or Maido (Japanese cuisine), Raphaels, or the Huaca Pucclana, which is a nice restaurant at the site of an pre-colonial adobe pyramid. I recommend you reserve ahead of time on line.
Taxi from downtown to Miraflores is approx 20 soles, and - depending on the traffic - about 30 minutes away. Again, leave downtown no later than 5pm, or you could have a one hour trip. Taxi back to your air bnb, if it is fairly close to the airport, would be approx 40 soles. The district of San Miguel is next to Callao (which is home to the airport). It is a good option for Air BnB, too. A little more upscale than Callao.