This is also not a complete answer, but I think technically the right track: Paul Broussous already suggested in his answer to define the Steinberg representation by induction of parabolics and Jim Humphreys pointed out that for general inclusions of Lie-Groupe there seem to be no easy branching rules (and suggested a small example, say $n=2$). I just want to tell you only what I see special about your case:
a) To be slightly more explicit on the first point: Let $I$ be a set of simple roots for $G$ $$St:=\sum_{J\subset I} (-1)^{|J|} Ind_{P_J}^{G}(1)$$ and this is related to to the similar expression for a Weyl group character, with $W_J\subset W$ the Weyl group generated by the simple elements longest element of the parabolic subsystem generated by the simple roots $J$: $$\epsilon:=\sum_{J\subset I} (-1)^{|J|} Ind_{W_J}^{W}(1)$$ One shows (Carter "Finite Groups of Lie Type", Chp. 6.2) that $\epsilon(w)=(-1)^{length(w)}$ and $\epsilon(w)=det(w)$ in the reflection representation.
Especially for $SL_{2n}=A_{2n-1}$ all $W_J$ are of type $A_k\times A_{k'}\cdots$ and $\epsilon$ is the sign-function in the symmetric group $\mathbb{S}_{2n}$.
b) A special relation between the Weyl groups / root systems / Tits buildings of $SL_{2n}=A_{2n-1}$ and $Sp_{2n}=C_n$ is as follows:
On $SL_{2n}$ you have an outer automorphism, say explicitly $f(X)=(X^T)^{-1}$. Correspondingly on $A_{2n-1}$ you have a diagram-automorphism $f$ of order $2$ and the orbits form a root system of type $C_n$.
Hence you can
- identify the $J'$-subsets for $Sp_{2n}$ with those $J$-subsets for $SL_{2n}$ which are $f$-stable; and especially the simple roots $\alpha_i',i\in I'$ of $Sp_{2n}$ with $f$-orbits $\{\alpha_i,\alpha_{2n-i}\}$ resp. $\{\alpha_n\}$ in $SL_{2n}$.
- Identify the Weyl group $W'\subset W$ of $Sp_{2n}\subset SL_{2n}$ to be generated by $s_is_{2n-i}$ and $s_n$.
- and so on...I think this is a good embedding of the Tits building in the sense of the comments above?
I am not sure this is enough to compute explicit branching rules for your question $$\chi:=Res^{SL_{2n}}_{Sp_{2n}}(St_{SL_{2n}})=\sum_{J\subset I} (-1)^{|J|} Res^{SL_{2n}}_{Sp_{2n}}(Ind_{P_J}^{SL_{2n}}(1))$$ i.e. to calculate $(\psi,\chi)$ for some irreps $\psi$ of $Sp_{2n}$. You might have a chance by starting with the principal series (maybe it's enough?), proceeding as in many proofs (e.g. $(St,St)=1$ in Carter Prop. 6.2.2), but I don't know an analogon to calculate double cosets $P_{J_1}'xP_{J_2}$ with $P_{J_1}'\subset Sp_{2n}$. At least on the level of Weyl groups computing double cosets $W_{J_1}'xW_{J_2}$ is definitely doable with the description in b), but I think this is not enough...