This paper reports a high-resolution experimental study of the $3^{3}\mathrm{\ensuremath{\Pi}}_{g}$ and $4^{3}\mathrm{\ensuremath{\Sigma}}_{g}^{+}$ electronic states of the $^{85}\mathrm{Rb}_{2}$ dimer. In the experiment, rovibrational levels of the two electronic states were probed using the perturbation facilitated optical-optical double resonance technique by exciting $^{85}\mathrm{Rb}_{2}$ molecules from thermally populated levels of the ground $X^{1}\mathrm{\ensuremath{\Sigma}}_{g}^{+}$ state through intermediate levels of the mixed $A^{1}\mathrm{\ensuremath{\Sigma}}_{u}^{+}\ensuremath{\sim}b^{3}\mathrm{\ensuremath{\Pi}}_{u}$ electronic states. The resonances of the probe laser were observed by detecting the laser induced fluorescence from the target states to the $a^{3}\mathrm{\ensuremath{\Sigma}}_{u}^{+}$ triplet ground state. In addition, to confirm the triplet character as well as the vibrational quantum number assignment of the states, for selected resonances the fluorescence to the $a^{3}\mathrm{\ensuremath{\Sigma}}_{u}^{+}$ state was resolved and bound-free spectra were recorded. From the observed term values for each state potential-energy curves were constructed using the Rydberg-Klein-Rees method.