Is this the actual expected cost of a price cap?

No. A real options-priced number needs a full caplet-strip model (the cap behaves like a series of caplets, one per reset) that this tool doesn't implement. This is a single-period lognormal expected-value estimate: it uses the index's own 12-month forecast as the forward price and its own historical volatility as the dispersion, and computes the probability-weighted excess over the cap (the same Black-76 formula used to price a call option), not a real-world-probability caplet price.