We provide an update on line intensity mapping surveys to be carried out with the CCAT-prime extreme field-of-view submillimeter telescope, which will begin operations at a 5600m site on Cerro Chajnantor overlooking the ALMA array in 2021. The first light survey with the EoR-Spec instrument module is anticipated to cover an area of 8 deg2, probing [CII] 158 micron emission at redshifts ~3.5-9.0, [OIII] 88 micron emission at redshifts beyond 7, and multiple CO lines covering all cosmic epochs where galaxies are known to exist. State-of-the art models suggest that the first light survey will yield detections of the [CII] signal to z>6 under the most conservative assumptions, and to z>8 under optimistic assumptions, achieving potentially as much as ten times higher signal-to-noise ratio as other contemporary measurements. Most of the survey area is anticipated to overlap with the HERA HI 21cm field, enabling cross-correlation between the distribution of the rest-frame far-infrared fine structure line emission from star-forming galaxies in the epoch of reionization and that of the neutral intergalactic medium. All data products from the surveys will be made publicly available to the community.