There are two ways to connect HDMI, and both will work on (just about) any case you buy.
1. If you use a separate video card then the HDMI cable plugs into the video card. The video card sticks out of the case through the PCIe slot. Cases have PCIe slots.
2. If you use integrated video provided by the CPU or MB chipset then the HDMI port is on the motherboard. The motherboard HDMI points sticks out of the case through a standard IO Shield. The case comes with a hole. The MB comes with an IO shield that goes in the hole. The HDMI port sticks out through the IO shield. Look at pictures of your MSI H110M Pro to see the HDMI port.
All this is hard to write, but very easy to see if you look at any youtube videos of building a PC.
Re choice of $50 case, this is more a personal "do i like the looks" exercise. One strong opinion: Do not get a case that comes with it's own junk power supply. Get a good power supply then get a case.
As a start, I like this case. Has USB3. Fractal Design Core 1100 Black MATX Mini Tower Computer Case. $28.99
http://www.newegg.com/Product/Product.aspx?Item=N82E16811352039&cm_re=micro_atx_case-_-11-352-039-_-Product