Neural-network models have been employed to predict the instantaneous flow close to the wall in a viscoelastic turbulent channel flow. Numerical simulation data at the wall is utilized to predict the instantaneous velocity-fluctuations and polymeric-stress-fluctuations at three different wall-normal positions in the buffer region. The ability of non-intrusive predictions has not been previously investigated in non-Newtonian turbulence. Our analysis shows that velocity-fluctuations are predicted well from wall measurements in viscoelastic turbulence. The models exhibit enhanced accuracy in predicting quantities of interest during the hibernation intervals, facilitating a deeper understanding of the underlying physics during low-drag events. The neural-network models also demonstrate a reasonably good accuracy in predicting polymeric-shear stress and the trace of the polymer stress at a given wall-normal location. This method could be used in flow control or when only wall information is available from experiments (for example, in opaque fluids). More importantly, only velocity and pressure information can be measured experimentally, while polymeric elongation and orientation cannot be directly measured despite their importance for turbulent dynamics. We therefore study the possibility to reconstruct the polymeric-stress fields from velocity or pressure measurements in viscoelastic turbulent flows. The results are promising but also underline that a lack of small scales in the input velocity fields can alter the rate of energy transfer from flow to polymers, affecting the prediction of the polymer-stress fluctuations. The present approach not only aids in extracting polymeric-stress information but also gives information about the link between polymeric-stress and velocity fields in viscoelastic turbulence.