We investigate the nature of dense gas in the 3─10 pc circumnuclear ring (CNR) in the galactic center of the Milky Way, which is a structure that may be dynamically connecting the supermassive black hole Sgr A* with the central molecular zone at the 100 pc scale, and is the closest reservoir of molecular gas to the massive stars located within the central cluster. In the first of several papers addressing open issues with the CNR, we use far-infrared (FIR) diagnostic emission lines to probe the hot and dense phase of the photodissociation region (PDR) exposed to the radiation field of the central population of massive stars. We use the Far Infrared Field-Imaging Line Spectrometer (FIFI-LS) instrument on board the Stratospheric Observatory For Infrared Astronomy airborne observatory to obtain spatially resolved maps of FIR emission lines of the region with an angular resolution approximately 4 times higher than previous published data. We complement our data with archival continuum images at 19.7, 31.5 and 37.1 μm obtained with FORCAST and 70, 100 and 160 μm archival continuum images from PACS. We use the FIFI-LS emission line flux maps from ionized ([C II] 157.7 μm), atomic ([O I] 63.2 μm, [O I] 145.5 μm), and molecular (CO J = 14─13 186.0 μm) species for a comparison with model predictions for PDRs. We present a method that dissects emission from the low and from the high excitation phase of the PDR and that also accounts for, e.g., absorption especially in the [O I] 63.2 μm transition. We present spatially resolved maps of dust temperature, atomic hydrogen column density, and FIR flux. The derived atomic hydrogen column density map is aligned with the galactic plane and extends spatially beyond previous near-infrared and radio based A v determinations. The atomic hydrogen column densities range from 1022.5 to 1023.1 cm−2 resulting in a total enclosed mass of the order of 103.5 M ☉. We derive a [O I] 63.2 μm absorption map that is aligned with the galactic plane with no or little absorption in the northern lobe of the CNR but moderate absorption in the southern lobe of the CNR, which is consistent with the picture where the illuminated front surfaces of gas clouds in the northern lobe are directly visible to us, while in the southern lobe the illuminated surfaces are hidden by the clouds within the lobe itself. Local gas densities in the CNR are generally below the Roche limit.