We study the behaviour of null-recurrent random walks on $\Z^+$ with a local drift that decays asymptotically in the distance $k$ from the origin with a power law with exponent $\alpha$ and an amplitude $b_k$. For the same models we investigate the meeting probabilities of independent random walks and use duality to apply these results to study aging in the Glauber-Isingspin relaxation dynamics. (AU)