Grating-based X-ray phase-contrast (gbPC) is known to provide significant benefits for biomedical imaging. To investigate these benefits, a high-sensitivity gbPC micro-CT setup for small (≈ 5 cm) biological samples has been constructed. Unfortunately, high differential-phase sensitivity leads to an increased magnitude of data processing artifacts, limiting the quality of tomographic reconstructions. Most importantly, processing of phase-stepping data with incorrect stepping positions can introduce artifacts resembling Moiré fringes to the projections. Additionally, the focal spot size of the X-ray source limits resolution of tomograms. Here we present a set of algorithms to minimize artifacts, increase resolution and improve visual impression of projections and tomograms from the examined setup. We assessed two algorithms for artifact reduction: Firstly, a correction algorithm exploiting correlations of the artifacts and differential-phase data was developed and tested. Artifacts were reliably removed without compromising image data. Secondly, we implemented a new algorithm for flatfield selection, which was shown to exclude flat-fields with strong artifacts. Both procedures successfully improved image quality of projections and tomograms. Deconvolution of all projections of a CT scan can minimize blurring introduced by the finite size of the X-ray source focal spot. Application of the Richardson-Lucy deconvolution algorithm to gbPC-CT projections resulted in an improved resolution of phase-contrast tomograms. Additionally, we found that nearest-neighbor interpolation of projections can improve the visual impression of very small features in phase-contrast tomograms. In conclusion, we achieved an increase in image resolution and quality for the investigated setup, which may lead to an improved detection of very small sample features, thereby maximizing the setup's utility.