We present IRAM 30m sensitive upper limits on CO emission in the ram pressure stripped dwarf Virgo galaxy IC3418 and in a few positions covering H ii regions in its prominent 17 kpc UV/H \alpha gas-stripped tail . In the central few arcseconds of the galaxy , we report a possible marginal detection of about 1 \times 10 ^ { 6 } ~ { } M _ { \odot } of molecular gas ( assuming a Galactic CO-to-H _ { 2 } conversion factor ) that could correspond to a surviving nuclear gas reservoir . We estimate that there is less molecular gas in the main body of IC3418 , by at least a factor of 20 , than would be expected from the pre-quenching UV-based star formation rate assuming the typical gas depletion timescale of 2 Gyr . Given the lack of star formation in the main body , we think the H _ { 2 } -deficiency is real , although some of it may also arise from a higher CO-to-H _ { 2 } factor typical in low-metallicity , low-mass galaxies . The presence of H ii regions in the tail of IC3418 suggests that there must be some dense gas ; however , only upper limits of < 1 \times 10 ^ { 6 } ~ { } M _ { \odot } were found in the three observed points in the outer tail . This yields an upper limit on the molecular gas content of the whole tail < 1 \times 10 ^ { 7 } ~ { } M _ { \odot } , which is an amount similar to the estimates from the observed star formation rate over the tail . We also present strong upper limits on the X-ray emission of the stripped gas in IC3418 from a new Chandra observation . The measured X-ray luminosity of the IC3418 tail is about 280 times lower than that of ESO 137-001 , a spiral galaxy in a more distant cluster with a prominent ram pressure stripped tail . Non-detection of any diffuse X-ray emission in the IC3418 tail may be due to a low gas content in the tail associated with its advanced evolutionary state and/or due to a rather low thermal pressure of the surrounding intra-cluster medium .