  PSR J1813-1749 is one of the most energetic rotation-powered pulsars known,
producing a pulsar wind nebula (PWN) and gamma-ray and TeV emission, but whose
spin period is only measurable in X-ray. We present analysis of two Chandra
datasets that are separated by more than ten years and recent NICER data. The
long baseline of the Chandra data allows us to derive a pulsar proper motion
mu_R.A.=-(0.067"+/-0.010") yr^-1 and mu_decl.=-(0.014"+/-0.007") yr^-1 and
velocity v_perp~900-1600 km/s (assuming a distance d=3-5 kpc), although we
cannot exclude a contribution to the change in measured pulsar position due to
a change in brightness structure of the PWN very near the pulsar. We model the
PWN and pulsar spectra using an absorbed power law and obtain best-fit
absorption NH=(13.1+/-0.9)x10^22 cm^-2, photon index Gamma=1.5+/-0.1, and
0.3-10 keV luminosity Lx~5.4x10^34 erg/s (d/5 kpc)^2 for the PWN and
Gamma=1.2+/-0.1 and Lx~9.3x10^33 erg/s (d/5 kpc)^2 for PSR J1813-1749. These
values do not change between the 2006 and 2016 observations. We use NICER
observations from 2019 to obtain a timing model of PSR J1813-1749, with spin
frequency nu=22.35 Hz and spin frequency time derivative
nudot=(-6.428+/-0.003)x10^-11 Hz/s. We also fit nu measurements from 2009-2012
and our 2019 value and find a long-term spin-down rate
nudot=(-6.3445+/-0.0004)x10^-11 Hz/s. We speculate that the difference in
spin-down rates is due to glitch activity or emission mode switching.
