You may look into literature where molar heat capacity of this compound might have been reported based on experimental data. If no data is reported, you can get first approximate values using Neumann-Kopp rule wherein the molar heat capacity of the constituents (here TiC and TiN) are added stoichiometrically. Data on TiC and TiN are available in literature.
First, please look into literature whether Enthalpy of formation of carbonitride has been determined experimentally. This is generally determined by bomb calorimetry. If you want to estimate the values (in case it is not reported already), you may consider TiC and TiN to for ideal solid solution. With this assumption, you may stoichiometrically add the enthalpy of formation of TiC and TiN. (As you know for ideal solutions the enthalpy of solution is zero).
There is a reference to this assumption and estimation of gibbs energy of formation: phys. stat. sol. (b) 209, 267 (1998)
For ideal solution, similarly you can estimate the entropy at 298 when you know S298 of TiC and TiN. Entropy of mixing is also to be added.