How to tell the polarization (horizontal or vertical) of a microstrip linear array antenna just by inspection
The short answer is yes, the position of the feed of a single patch determines the horizontal vs. vertical polarization of that patch.

Since all the patches in the linear array are fed in the same way, this determines also the polarization of the array.

See this article, which covers the working of single-patch antennas:


The Basics of Patch Antennas, Updated; By D. Orban and G.J.K. Moernaut; Orban Microwave Products


Some excerpts:


  Topics include principles of operation, impedance matching, radiation patterns, circular polarization, bandwidth, efficiency, alternative feed types, stacked patches and higher mode behavior.


[...]

The relevant section for your question is the following:


  Polarization
  
  The plane in which the electric field varies is also
  known as the polarization plane. The basic patch covered  so far is
  linearly polarized since the electric field varies in only one
  direction. This polarization can be  anything between vertical and
  horizontal depending on the orientation of the patch. The polarization
  plane  is the xz-plane in Figure 1. For optimum system performance,
  transmit and receive antennas must have  the same polarization. The
  patch described above yields horizontal polarization and when rotated
  by 90°,  the current flows in the vertical plane and the antenna is
  now vertically polarized.


[...]


  


The previous image, which in the article is used to illustrate how to obtain a circular polarization, explains why the feed position determines the polarization: feed1 and feed2 excite different linearly polarized modes (TM10 and TM01), which combine to give circular polarization. From that it is clear (read the text in the article) that if only one feed is present, it excites just one linearly polarized mode.