I understand why time experienced by someone travelling close to the speed of light runs slow (in relative terms) due to the increase of mass (?) however, I am having trouble finding an answer to the following question.

Imagine a hypothetical means by which a person can travel (near instantaneously) from point A to point C without crossing point B (the intervening distance) - whether "wormhole" or "space warp", the method of travel does not matter in this instance. Anyway, if points A and C are one light year apart, then the journey is surely analogous to FTL (which is as we know impossible). My question is this: Given that this hypothetical journey involves no object literally travelling at any particular velocity, and thus not accumulating additional mass, would time (as experienced by this traveller) run at the same rate relative to that of a non-traveller (irrespective of what happens to the light) or would some dilation (in theory) occur? Would the journey be truly instantaneous or would the person arrive at point C some great length of time before or after departure from point A (keeping in mind the fictitious nature of this form of travel).

Apologies for what may be an awkward (and first!) question (as in poorly expressed), or what I realise may not even be a legitimate question. Any advice or pointers would be greatly appreciated. I'm actually a (fiction) writer so this is research for something I'm working on. I think it's important (for the purposes of the question) to ignore the impossibility of FTL travel or indeed my suggested equivalent-without-physical motion alternative to FTL travel. I suppose the core elements of my question are:

1) Increased mass (leaving aside violation of causality as an"after effect") prevents FTL travel, yes? Where does this mass derive from, or rather which part of the travel (I'm assuming it's the velocity) leads to the increase of mass?

2) If (hypothetically speaking) one could travel close to the speed of light without an increase in mass, would FTL then be (in theory) possible (once again suspending disbelief where causality is concerned)?

Ugh. Think I need to sit down...

PS. If any mod thinks there might be a more appropriate section for this question, please feel free to move it. I'm new here so I'm bound to bump into a few things at first.