  The characteristic age of PSR J1734$-$3333 estimated from its current spin
down rate implies that it is a young pulsar ($\tau_c<10$ kyr). But the time
derivative of its spin down rate differs markedly from that assumed for normal
radio pulsars, meaning its actual age is uncertain. G354.8$-$0.8 is a supernova
remnant (SNR) whose centre is located 21 arcmin away of the pulsar, and with a
morphology that suggests an association with the pulsar. We want to assess the
likelihood of the association between PSR J1734$-$3333 and G354.8$-$0.8 or
other nearby supernova remnants quantitatively, with the objective of shedding
light on the real age of this pulsar. Observations with the Karl G. Jansky Very
Large Array were carried out in 2015 and 2019 that allow precise astrometric
measurements and consequently a proper motion estimate for the pulsar. The
proper motion was found to be $\mu_\alpha=10\pm10$ mas yr$^{-1}$ and
$\mu_\delta=-29\pm11$ mas yr$^{-1}$ (error bars are $1$-$\sigma$). Though
marginal, this detection rules out the association with G354.8$-$0.8 because it
means the pulsar is not moving away from the centre of the SNR. No SNR
consistent with the measured proper motion and an age $\sim\tau_c$ could be
found. We also present the first measurement of the spectral index for this
pulsar, $\alpha=-1.1\pm0.3$, measured between $1.5$ and $3.0$ GHz. The SNR
produced by the birth supernova of PSR J1734$-$3333 could have already faded to
undetectable brightness, for which estimates suggest timescales of $10$-$100$
kyr. This and other considerations lead us to conclude that the pulsar is
possibly older than $45$-$100$ kyr. PSR J1734$-$3333 is a pulsar with
rotational properties that place it between standard radio pulsars and
magnetars, and we interpret our result in the context of a possible future life
as a magnetar for this pulsar.
