Can this Fritzing diagram be simplified to share a single hole?
Each column of five holes is one metal strip. You can confirm this with a multimeter or continuity tester.


  It seems to me that I can just combine all 4 legs into the same hole? (the + side of the capacitor, 1 leg of both resistors, the 1 screw terminal)?


Not the same hole; into the same strip. If the strip hasn't enough holes you can jumper out to another strip as is done with the yellow jumper.


  Am I wrong in this? I watched a youtube video on them building this circuit and they indeed did different holes and bridged them together with solder.


You don't use solder on a solderless breadboard. You were looking at a solder perforated board.



Now that the question is cleared up ...



Figure 1. The "single-wire-per-hole" strategy will make it easier to remove components for circuit modification. The "stuff-them-all-in-one" strategy will make alterations much more difficult but may make tracing easier as it will be obvious what is connected.

I've only used that type of board a couple of times and found it very difficult to make a good looking job. The standard strip-board or Veroboard is much simpler to use and very versatile. 



Figure 2. Rather neat circuits can be built on Veroboard if the components can be mounted perpendicularly to the direction of the copper tracks. Here there are no flying wires and all cross-connections are done with components or wire links. Image: Wikimedia Commons.



Figure 3. A Vero tool is rotated by hand to remove copper around any hole thereby isolating the strips either side of that hole. This tool doesn't go right through the board as in Figure 1.

Other options such as solderless breadboard patterned versions exist also.



Figure 4. The breadboard-patterned version. Source: Wikimedia Commons.

There are many stripboard designers available which allow you to place components, links and break tracks. These can help to generate a plan for a very neat layout.