The gas-rich protoplanetary disks around nearby young stars are regions of active planet formation. Recently, the presence of young planets in protoplanetary disks has been inferred by identifying deviations from Keplerian rotation in spatially resolved maps of spectral line emission. Debris disks around slightly older main sequence stars occasionally have sufficient quantities of gas to allow for a similar study of disk kinematics at comparable spatial and spectral resolution. Here we apply the methodology used to infer the presence of planets in protoplanetary disks — via signatures like velocity kinks and Doppler flips — to the debris disk around the nearby A star 49 Ceti. We search for localized velocity kinks within channel maps and Doppler Flips in the residuals of intensity-weighted velocity (first moment) maps. Using archival data from the Atacama Large Millimeter/submillimeter Array (ALMA), we first verified the presence of velocity kinks and Doppler Flips in the protoplanetary disk around HD 163296, using 12CO(J=2-1) at an angular resolution 0.04” (linear resolution 5 au). We then analyzed CI(3P1-3P0) (angular resolution 0.25”, linear resolution 14 au) and 12CO(J=3-2) emission data (angular resolution 0.13”, linear resolution 8 au) for the gas-rich debris disk of 49 Ceti, which did not show any clear velocity kinks/wiggles. However, disk models do indicate moderate evidence of a warp in the disk, a possibility considered in previous studies. Our analysis rules out gas giant planets orbiting 49 Ceti with semimajor axes between 60 and 260 au, and provides proof of concept for similar studies of a wider variety of debris disks in the future.