I would imagine that the substitution would be faster as you might get some anchimeric assistance from the nearby oxygen.
When you said tosyl (and in your image), did you mean that it is an O-Ts group, or just a tosyl connected to the alkyl chain (which I'm not familiar with)?
If it's the first, I'm curious sjb why the O-Ts group won't be kicked out and replaced by the OMe group, as the anion that is kicked out is relatively stabilized and not likely to react further.
How would the methyl tosylate be generated?
I wish I could explain better but I'm on a friends computer and don't really want to "paint" draw what I'm trying to say.
Sorry, hopefully I haven't confused the issue...