Type radiation used carbon dating
In this case, depending on the quality of the instrument (the energy resolution), you should basically be able to determine which types of radiation are present in your sample's energy spectrum.
What's going to happen after one billion years? And then you didn't build your time machine well.
That's about all you really need, because C-14 decays fast enough that it's not really useful past about 50,000 years.
C-14 decays by beta emission, which occurs with basically a continuum of energies up to 149 ke V (link).
This isn't entirely true, as higher energy gammas will produce some background at lower energies due to compton scattering, but in theory you could subtract this out.
If you had contaminants that were low energy gamma emitters (less than 149 ke V), or other beta emitters (which will also produce a spectrum of energies up to some cut-off value), this will complicate things more worser.