We present a chemical abundance study of giant stars in the Galactic open cluster NGC 5822, which hosts two barium stars (#002 and #201) and three lithium-enriched giants (#006, #102, and #240). Using high-resolution optical and near-infrared ($H$ and $K$ band) spectra from FEROS and IGRINS, we determine atmospheric parameters and abundances for 23 elements (Li, C, N, O, F, Na, Mg, Al, Si, P, S, K, Ca, Sc, Ti, Cr, Fe, Ni, Y, Ce, Nd, Yb, and Pb). This includes species not yet studied in this cluster, such as F, P, K, Yb, and Pb, as well as oxygen isotopic ratios $^{16}$O/$^{17}$O and $^{16}$O/$^{18}$O. Membership was assessed using astrometry and chemical abundances, providing insight into the evolutionary stages of Li-enriched giants and cluster parameters (age, distance, extinction). However, the identification of Ba-stars remains challenging due to their binary nature and less reliable astrometric solutions. The cluster's abundances are broadly consistent with expectations for the Galactic thin disk. The mean fluorine abundance agrees with chemical evolution models predicting that young clusters (<2 Gyr) exhibit elevated [F/Fe], with production from SN II, SN Ia, AGB, and Wolf-Rayet stars. No distinct chemical or rotational features were found to explain the lithium enrichment, likely occurring either during the red clump phase or near the RGB tip. For the Ba-stars, nucleosynthesis models combined with the cluster's turn-off mass suggest polluting companion masses of 3.00 and 3.75 $M_{\odot}$ for stars #002 and #201. These results highlight the importance of open clusters as laboratories for chemically peculiar stars.