Is this planned building in a runway approach legal?
If it falls within the limits set out by FAA FAR Part 77 then its legal. You can find a nice breakdown of Part 77 here but you would need measurements more precise than what google maps is going to give you to be sure. Depending on the height the FAA may also require them to light the top of it. This is actually an interesting break down as the FAA is a federal organization and building approval generally comes down at the city/town/municipality level. If a building is approved for construction by a city one can assume it is legal to be built and that the city accounted for the FAA regulations. According to AC 150/5190-4A which quotes the Airport an Airway Improvement Act of 1982 it looks like its on the city to ensure that nothing gets in the approach path: 


  ...Sec. 511 (a) SPONSORSHIP. As a condition precedent to approval of an
  airport development project contained in a project grant application
  submittted under this title, the Secretary shall receive assurances in
  writing, satisfactory to the Secretary that ... (4) the aerial
  approaches to the airport will be adequately cleared and protected by
  removing, lowering, relocating, marking, or lighting or mitigating
  existing airport hazards and by preventing the establishment or
  creation of future airport hazards; (5) appropriate action, including
  the adoption of zoning laws has been or will be taken, to the extent
  reasonable, to restrict the use of land adjacent to or in the
  immediate vicinity of the airport to activities and purposes
  compatible with normal airport operations, including landing and
  takeoff aircraft; ...."


In other words, the city needs to have zoning laws that keep the approach path clear. This is largely covered under the above linked Part 77 but generally speaking if the building has been approved its more than likely legal as per approved zoning laws. 

A bit of a related story: you should not put it past people to build out of spite. A farmer once built a silo right off the end of Chatham Airport's runway to try and stifle traffic that he claimed was buzzing him while he was working. The silo is now gone and I cant find any info on what happened to it but this was also some time ago and I dont know what the regulations were when it happened. 

And in some rare cases, if there is something in your way, an approach can be devised around it.