Okay, download the patcher and then unzip it to a directory on your computer.
Right click on wherever the file downloaded to and press "extract all". That will extract the files and place the extracted version right where you saved the zip file to. Thus, the zip file should be in that folder/directory as well as the new unzipped files. These are "sodbin.exe", "sodpatcher.exe" and "filelist" or something along the lines of "filelist".
Now double click on the patcher and it should ask you to select your EQ directory (if not, when you open the patcher, hit "advanced" and then press "select EQ folder" (or something along those lines)). Now you'll have to select your EQ folder.
By default, your EQ folder should be: c:\Program Files\EverQuest
Select that by manually going to it, then when you have EverQuest highlighted, press okay. Everything should be working now.
What I think you did wrong was by not selecting your actual EQ directory, but instead something else. It started patching, then realized there was no eq.dll and stopped.