PSR J0837$-$2454 is a young 629 ms radio pulsar whose uncertain distance has important implications. A large distance would place the pulsar far out of the Galactic plane and suggest it is the result of a runaway star, while a short distance would mean the pulsar is extraordinarily cold. Here we present further radio observations and the first deep X-ray observation of PSR J0837$-$2454. Data from the Parkes Murriyang telescope show flux variations over short and long timescales and also yield an updated timing model, while the position and proper motion (and, less strongly, parallax) of the pulsar are constrained by a number of low-significance detections with the Very Long Baseline Array. XMM-Newton data enable detection of X-ray pulsations for the first time from this pulsar and yield a spectrum that is thermal and blackbody-like, with a cool blackbody temperature $\approx$$70\ \mbox{eV}$ or atmosphere temperature $\approx$$50\ \mbox{eV}$, as well as a small hotspot. The spectrum also indicates the pulsar is at a small distance of $\lesssim$$1\ \mbox{kpc}$, which is compatible with the marginal VLBA parallax constraint that favours a distance of $\gtrsim$330 pc. The low implied luminosity ($\sim7.6\times10^{31}\mbox{erg\, s}^{-1}$ at 0.9 kpc) suggests PSR J0837$-$2454 has a mass high enough that fast neutrino emission from direct Urca reactions operates in this young star and points to a nuclear equation of state that allows for direct Urca reactions at the highest densities present in neutron star cores.