What are the molecular orbitals shaped like in diberyllium (mono)cation? I think one way of describing the result is that the bond in $\ce{Be^+_2}$? cation is a resonance hybrid of two possible pi bonds. Each pi bond has a bond order of 1 so the resonance hybrid structure has two partial pi bonds. But then can it further be in a resonance hybrid of different rotations and so be an even cylinder?

I guess if you wanted to be silly you could say that that the electron configuration for each beryllium ion is like:

$$ \underset{2s}{[\uparrow \downarrow]} \underset{(2p)^2}{[\uparrow \vert \;]} \underset{2p}{[\;]} $$
Short answer: the molecular orbitals will be $\sigma$ and $\sigma^*$ and thus cylindrically symmetric.

First off, your electron configuration is wrong. $\ce{Be2+}$ will have 3 electrons, so the $\ce{2p}$ orbitals need not be considered.

Here's a quick MO diagram:



So there aren't an $\pi$ bonds at all. There's a $\sigma$ and a $\sigma^*$ orbital, for a bond order of 0.5.

If I generate orbitals (using Avogadro) then I see something like this for the $\sigma$



And this is the $\sigma^*$