Guide on how to connect, geared towards PCSX2 users (credit to Haze1079):
You need the CLR_DEV9 plugin (CLR DEV9)
You need an ethernet plug (as far as I'm aware)
Keep all settings of CTR Dev9 Automatic
Skip to step #5 or step #6 if your playing on a PS2 
If you downloaded Kori's PCXS2 Setup, or if you already have your NETWORK File configured for this server, skip to step #37 and choose the "Previous Connection Option" Option 2.
#0 Boot up PCSX2

#0.1 Go to the Config toolbar near the top of the window and click on it

#0.2 Click on the plugin/BIOS Selector

#1 Use the dev9 plugin listed in the Plugins settings

#2 Make sure you have the proper BIOS selected (JP v01.60 (around 2004 time). I was using a USA v01.60 BIOS (around 2002))

#3 Have a physical copy of MH1 JP PS2

(Optional: Download the English Patch for MH1 JP PS2)

#4 Boot up the game with the proper BIOS

   (If you already have a character skip number 5 and number 6)

#5 Make an offline character

#6 After that is done, save and quit with your new character

#7 Go to the continue screen, choose the memory card that contains the character that you want to go online with.

#8 Choose your character, and then when given the option to go to Village or Town, go to Town

#9 When given the option in the NETWORK.ONLINE screen to load NET data from your memory card keep clicking circle
#10 Once you get to the screen where you see three options (the bottom two should be blacked out) go to the top option and click it
#11 When presented with the subsequent menu click the top button

#12 Click the top option given in the next menu

#13 Then click the top option of that menu

Now you should be in a screen with three blank options it is black and blue and the uses of each line is displayed in the top-left
corner. If you don't have an English Patch just follow context clues.

#14 Pick the very top option

#15 Choose the memory card that contains the character that you want to play online with

#16 When the "Setting Guide Mode" screen appears (the first screen) click right on the Dpad
#17 On the "Hardware Settings" page click circle (there should only be one option)

#18 Now click right on the Dpad again to continue

#19 Click on the bottom option of the "Use of PPPoE" screen

#20 Click right on the Dpad

#21 On the "IP Address Setup" page click right on the Dpad (as it should be set to automatic by default)

#22 On the "DNS Server Address Setup" choose the bottom option (as you don't want it to be automatic)
#23 After selecting click right on the Dpad to go to the "DNS Server Address Entry" page

#24 Click circle to enter numbers on this page

#25 You may have to count the numbers manually for the first entry, but make sure it is 185.219.132.136 (the server's current address)

#26 It may say that the number didn't save in the top left corner of the screen (by indicating a different number) but it did.

#27 Now go to the bottom option on the "DNS Server Address Setup" page and repeat what you did in step #25

#28 After doing that click circle to confirm then right on the Dpad

#29 In the "Name the Setting" page click right on the dpad as you can omit it

#30 On the "Confirm and Save the Setting" page double check that everything being displayed in the top right corner is correct
    (Hardware, Mac Address, and Hardware Operation Mode shouldn't matter. IP Address should be Auto, and both Primary and Secondary
    DNS should be what you typed in, 185.219.132.136 (the server's current address))

#31 Now once everything is ok click circle again

#32 Once it is done saving, when given the option to test the connection click the right option, which is "No".
#33 Once you return to the network settings menu click x and quit it by selecting the left option (DO NOT REMOVE YOUR MEMORY CARD
    DURING THIS ACTION)
#34 Now click continue and load the memory card that contains your character

#35 Load your character and head online to the Town

#36 Keep clicking circle until you get to the menu with the three options

#37 Choose the top option (first time) (subsequent times you can choose the "Previous Connection Option" Option 2)

#38 When brought up with the options menu that brought you to step #14, you should see a new option at the bottom of the options

#39 Scroll to it and click circle on it

#40 Voila, now if everything went to plan it should connect you to the server

#41 Click circle to agree to the EULA

#42 When brought up with the "Monster Hunter private fan server" page create a new account with whatever name and pass you want. 
    (or use an existing one if you have done this before)
    DONT SHARE THIS AS THERE'S NO WAY TO GET YOUR ACCOUNT CHANGED. 
    Write down your login info.