EDIT: After I posted my answer, I realized that you were very close to answering your own question---the vector bundle you want is the one whose fiber over $\alpha \in {\rm Jac}(C)$ is $H^{0}(\mathcal{O}(2\Theta) \otimes \alpha)^{\vee}.$ However, this is equivalent via Strange Duality to what I have written below.

In general, if $C$ is a smooth projective curve of genus $g \geq 2$ $U_{C}(r,0)$ is the moduli space of S-equivalence classes of vector bundles of rank $r$ and degree $0$ on $C,$ and we fix a line bundle $L$ on $C$ of degree $g-1,$ the locus
$$\Theta_{L} = \{E \in U_{C}(r,0) : h^{0}(E \otimes L) > 0 \}$$
is an ample divisor on $U_{C}(r,0).$ Recall that the fibers of the determinant map ${\rm det} : U_{C}(r,0) \to {\rm Jac}(C)$ are each isomorphic to ${\rm SU}_{C}(r)$, which parametrizes rank-$r$ vector bundles with trivial determinant. For any two line bundles $L,L'$ of degree $g-1$ on $C,$ the restrictions of $\Theta_L$ and $\Theta_{L'}$ to any fiber of ${\rm det}$ are linearly equivalent; the resulting ample line bundle on ${\rm SU}_{C}(r)$, which is denoted by $\mathcal{L},$ is called the $\textit{determinant bundle},$ and it generates ${\rm Pic}(SU_{C}(r)).$

For $k \geq 1$, the $\textit{Verlinde bundle}$ on the Jacobian ${\rm Jac}(C)$ corresponding to the rank-level pair $(r,k)$ (and our choice of $L$, typically a theta-characteristic) is the vector bundle $$\mathbf{E}_{r,k} := {\rm det}_{\ast}\mathcal{O}(k\Theta_{L})$$

For each $\alpha \in {\rm Jac}(C)$ the fiber of $\mathbf{E}_{r,k}$ over $\alpha$ is isomorphic to
$$H^{0}({\rm SU}_{C}(r,\alpha),\Theta_{L}|_{{\rm SU}_{C}(r,\alpha)}) \cong H^{0}({\rm SU}_{C}(r),\mathcal{L}).$$
In the case where $g=r=2,$ the paper of Narasimhan-Ramanan that you are reading proves ${\rm SU}_{C}(2) \cong \mathbb{P}^{3}.$ In particular, the determinant bundle $\mathcal{L}$ on ${\rm SU}_{C}(2)$ is exactly the hyperplane bundle $\mathcal{O}(1)$ on $\mathbb{P}^{3}.$ The map ${\rm det}$ then realizes $U_{C}(2,0)$ as a $\mathbb{P}^3$-bundle over ${\rm Jac}(C)$; this is the projectivization of the rank-4 Verlinde bundle $\mathbf{E}_{2,1}.$