The answer to your question has to deal with how electronegative hydrogen is. Even though hydrogen has 1 valence electron like the alkali metals, hydrogen is a small atom and does not experience a shielding effect. Also, since that 1 electron is really close, it is harder for hydrogen to lose that electron than it is for alkali (and all other) metals. To prove my point, hydrogen has an electronegativity value of 2.1. The highest electronegativity value a metal has is 2.0 (gold). The lower the electronegativity value is, the more readily and easily an element will give up its electron(s). For something to become oxidized, it must be able to readily give up electrons. Finally, it's really unrealistic to use solid hydrogen in a cell. You need to achieve a temperature of 14.01 K in order to make hydrogen a solid. Since you need an ionized liquid with a salt in it (metal on the cathode + something else), it would be impossible at 14.01 K assuming standard temperature pressure.
tl;dr
-hydrogen has a greater pull on its electrons because there is less distance between the nucleus and the electron
-only 1 electron, therefore there is no shielding effect (which big elements have)
-results in a higher electronegativity than all metals
-too difficult with temperatures, cell probably wouldn't even work correctly