I think the 'Expert' tag is a little misleading. I struggled heavily with the 'normal' challenges first but once you figure out the logic behind each challenge the expert ones are not really that different. So that would be my first piece of advice - disassemble and analyse the board state after every turn and try to plan for your win condition. Analyse what conditions you would need to win then work backwards to your starting hand. Your starting cards, your enemy's hand, and the general order they play it in never change for the first two challenges.
The first challenge there is no variance whatsoever. The idea is since the effect of your archers is scalar you want to play multiple archers consecutively. Since your hero effectively takes a turn to invert your current units you'll want to use a burst of archers to completely clear the field then finish off with a burst of scythemen. Try and time Germain so he comes out just before a katakan comes out to kill a peasant.
Second challenge is just a more complicated setup. Start off with Germain for tempo then line up waves of vampires with archers to finish them off with Vesemir, and try to time Vesemir so you can use him a second time when the first katakan comes out.
For the final challenge with Regis, where the starting conditions are less predictable, you can still set yourself your own conditions. You don't have a full deck so your starting hand is generally consistent. Since your peasants have exponential effects try making a big play to win the first round, pass on second round where the rats also thin from your deck, then use everything else you got to win round 3. I like to try and keep the crossbowmen for round 3.