The physical origin of fast radio bursts (FRBs) remains uncertain. Although multiwavelength observations have been widely conducted, only Galactic FRB 20200428D is associated with an X-ray burst from the magnetar SGR J1935+2154. Here we present multiwavelength follow-up observations of the nearby bright FRB 20250316A, including the Five-hundred-meter Aperture Spherical radio Telescope (FAST), Einstein Probe (EP) X-ray mission, Chandra X-ray Observatory, Wide Field Survey Telescope (WFST), and Space Variable Objects Monitor/Visible Telescope (SVOM/VT). The 13.08 hr FAST follow-up campaign without pulse detection requires an energy distribution flatter than those of well-known repeating FRBs, suggesting that this burst is likely a one-off event. A prompt EP follow-up and multiepoch observational campaign totaling >100 ks led to the detection of an X-ray source within the angular resolution of its Follow-up X-ray Telescope (FXT; 10 ''). A subsequent Chandra observation revealed this source to be offset by 7 '' from the FRB position and established a 0.5-10 keV flux upper limit of 7.6 x 10(-15) erg cm(-2) s(-1) at the FRB position, corresponding to similar to 10(39) erg s(-1) at the 40 Mpc distance of the host galaxy NGC 4141. These results set one of the most stringent limits on X-ray emission from a nonrepeating FRB, disfavoring ultraluminous X-ray sources as counterparts of apparently one-off FRBs and offering critical insights into afterglow models. Our study suggests that an arcsecond localization of both the FRB and its potential X-ray counterpart is essential for exploring the X-ray counterpart of an FRB.