The Cub instructor can do it. Stay in the pattern with him/her for five hours. You will be ok.
They told me that if I learned to fly the Cub well I could fly anything. That is close to true - while 20 Cub hours is not enough, your instructor probably has several hundred. I had no problems checking out in other taildraggers - sometimes nobody is around who is qualified, so I have to carefully do a self-checkout. Not optimum, but sometimes necessary. From memory, my self checkouts were C-180, UPF-7, V-77 Stinson, 450 Stearman, Super Cub, 108 Stinson, Aeronca Chief, and J4/J5. I always get a checkout from a knowledgable instructor when I can.