There are several reasons why $\mathrm{^{13}C}$ experiments are far less sensitive than proton experiments.
The natural abundance for $\mathrm{^{13}C}$ is only 1.1%, while $\mathrm{^{1}H}$ has a natural abundance of 99,9%. This difference alone means that if you don't label your sample with $\mathrm{^{13}C}$, the spectra will be around 90 times less sensitive than proton spectra due to the lower natural abundance alone.
$\mathrm{^{13}C}$ also has a four times lower gyromagnetic ration than $\mathrm{^{1}H}$, which also lowers the sensitivity. But the way the experiment is usually measured, you get magnetization transfer from directly-attached protons to the carbons, which increases the sensitivity.
$\mathrm{^{13}C}$ also relaxes more slowly, which means that the time between experiments is longer and you can measure fewer scans than for proton experiments in the same time.
Another factor that is important is which kind of probe your spectrometer is using. For an inverse probe like e.g. a Bruker TXI, the inner coil is the proton coil, which means it's especially suited for proton experiments. For $\mathrm{^{13}C}$ experiments an observe probe like a BBO is more sensitive, as the carbon is measured on the inner coil. If you have multiple spectrometers, you should ask which one is better suited for $\mathrm{^{13}C}$ experiments.
If you have a very small amount of sample, a Shigemi tube allows you to measure in half the sample volume (200-250 $\mu$l) compared to standard tubes, you can achieve a higher concentration that way. They're pretty expensive though and they have to be matched to the solvent, so you would need a CDCl3 tube.
And there are various problems that could drastically reduce your sensitivity for $\mathrm{^{13}C}$, the most important ones if we disregard hardware problems, are proper tuning and matching for $\mathrm{^{13}C}$, and the correct pulse length for $\mathrm{^{13}C}$.
I've avoided your actual question, as the required concentration depends on quite a few factors. I'd recommend you to ask the person responsible for the spectrometers, as they should be able to give you a better estimate based on the actual hardware you have.