One day in rainy Bucharest

The forecasts say it's going to be cloudy, not necessarily rainy.

Are there any interesting exhibitions going on (or something similar) where I could spend half a day?

You can check a list of exhibitions open this week here.
You could also try the National Art Museum or the Contemporary Arts Museum.

can anyone tell me what is the best way to get to the airport from the city centre?

Express bus 783 (runs every ~15 minutes, costs €2 return).

Thanks. Yes, the forecast is better now than it was in the morning.