Make sure you have done a Reset of the iPod since this problem occurred
A Reset just restarts the iPod. It is not erased.
When you say the screen is "turned off," do you mean that then backlighting for the screen is not on, or that nothing appears on the screen (with or without backlighting)? If it's only the backlighting, it may be the setting. You can look at this setting on the iPod, on its Settings screen, for Backlight Timer. Make sure it is not set to Off.
If you have not already, you can try doing a Restore on the iPod using iTunes. The Restore button is on the iPod's Summary screen in iTunes. This will erase the iPod, reinstall its software, and set it to default settings. If the problem is related to software or settings, a Restore should fix it. If there is a hardware problem on the iPod, it will not help.
The FireWire docking cable should not be a problem, if it is able to connect the iPod to iTunes and charge it.
The backlight is 100% working, the only problem is that I don't see the letters (with backlight or not). Sometimes the letters appear but in one minute they become disordered and disappear for days and month, always. I keep listening to music but can not see anything. I reseted an restored it many times and nothing.
You said that is a hardware problem... well nobody kwows how to fix this kind of iPod in this country.
If you did a Reset and a Restore, then the problem is likely to be a hardware problem. Because it is any older iPod, it may not be economical to have it repaired, even if you can find a repair business that can perform the repair.
If you are handy with fixing things yourself, you can look here for do-it-yourself guides.
There are probably also YouTube videos on the topic. Compared to newer iPods, 4th gen (and older) iPods are easier to open and fix.
Most older iPods that have a hard drive are broken because of a faulty hard drive. Other common problems are with the battery, click wheel, dock connector, and headphones jack. Since your iPod is broken for a less common reason, you may be able to find and buy a broken iPod of the same type. Find one that has a good screen. Swap parts to make one working iPod.
What a bad luck... the most uncommon problem in my iPod (it have five hours of battery nine years later)... I'm good reparing things with my hands... but I don't like to repair Apple products... Years ago I fix my old iPod mini. Now is working but I left it with a lot of scratches in the aluminium and above plastic.
If I found an old and broken iPod Clickweel maybe. But that it's so difficult to found in my country.
I hear iPod mini is difficult to open without causing damage. The 4th gen iPod is easier to open without causing damage, if you use the plastic (nylon) tools shown in the guides. I hear that guitar picks also work for this purpose. Be careful along the left side of the iPod (looking at the iPod from the front); there is a delicate ribbon cable there that connects the two halves of the iPod's case. I say it's "easier," but it's still not easy for most people.
It's possible that there is an internal connection that has come loose inside. There is ribbom cable that connects the display to the logic board. You can see it in this repair guide
in Step 17. If that connection is loose (or has oxidation), maybe just removing it and reconnecting may work. You would not need to remove the logic board, since you are not taking out the display. If you feel like taking it apart, that would be something you can look at without the need for replacement parts.