The solo content is f2p in the sense that you do not need to pay money, but only ingame gold. It is possible to unlock it without paying.
However, it is not necessary to buy it for competetive reasons. You will most likely get no special cards that are better than the cards you get through packs. The maximum value usually depends on the specific prices and the rewards you get. For the last adventure it wasn't worth buying it with gold if yu do not like PvE content since it was overpriced if you only look at the costs and rewards in pack value (not too much though).
In conclusion, you do not need to buy the adventures since they do not give you comptetive cards (at least not yet).