Work with your landlord on this one to determine what can be done and what agreements are in place with Time Warner. It's ideal to have Fiber installed directly into your unit rather than to let Verizon set you up with a VDSL2 connection + Coax. In many cases, the coaxial lines in an apartment are all tied back to a central system, which is likely what Time Warner is using. Verizon can likely adapt to place the ONT near the location where the Cable lines terminate, but they still have to get the Fiber in place. They will not re-use conduit that Time Warner is using to get the Fiber in the building, and if the coaxial cabling is owned by Time Warner, they will not touch that either.
If your landlord can be on site during install day, have him or her work with the Verizon to try and determine the best way to run everything, or ideally in a way so that others can get FiOS easily without having to drill holes into the building in bad spots. Once they get the Fiber into your unit, the most they would need to do from that point is find a way to get power to everything, and to run or re-use coax.