Cool name, but this mission is seriously driving me insane.
I can scan everything, and fly by the docking bay of the Lucifer, but then I'm shot down by Rama as I try to jump out. I'm using countermeasures, flying in zigzags, using afterburners, focusing the shield in back, the works. Nothing helps. Is there some trick I'm missing? Is it possible to not be noticed?
Anyway, if anyone can tell me what I'm doing wrong, it would be greatly appreciated.

My technique is to put all power to engines and race in to scan things and then race out again... And keep repeating.
And then flee as fast as possible at the end.
Karl

when all else fails, lower the difficulty.

Rise engines, leave shield power while draining weapons (wich you wont be using). DON'T LET A SINGLE SHOT OUT or you will be discovered!
start scanning cargo 1 & 2, then the freighters, then cargo 3 & 4 for the bonus objective. during this pause monitor arjuna wing so you can stay out of their flight path. repeat scans and monitoring with every freighter, Eva & the Lucifer's fighterbay (I think it was fighterbay 1).
Then you need to hold the horde of fighters, and at the same time evade the capital ships. Try boosting out of the ships blasters, then reset your energy configuration and blast some of the fighters till your subspace drive comes back online
Its one of the hardest Freespace 1 missions, be ready to suffer through it.

Hell, I find it difficult to get close to the things needed to scan them without the shivans getting wind. I'm feeling that I should just blow this mission and move on.

Pump all your power into engines, fly up to mobile targets as soon as they come out of the jump gate, and deal with the cargo when you have a free moment. If you get a warning, turn around and fire your afterburners away from red dots. Pray.

Followed this to the letter and I only got two warnings, I scanned everything and legged it quick when I was told to jump back after completion ^_^

Right at the start I also just shove all power to engines and head directly for the cargo containers. You can usually scan one before the first two ships jump in, fly directly too them and scan 'em using the speed matching thing to simplify things.
You should then be able to get another two cargo containers, even the rest. Before the next lot jump in.
I found that as long as you stay around the area that the ships jump into you should be able to get the mission done easily

Is it a big deal if you fail this mission and move on?
I got frustrated by this mission and ended up accepting a Fail. I skipped scanning the cargo and freighters. I got most of the capital ships and then I missed scanning one of the cruisers before it jumped out.
I managed to escape after I was discovered, and decided to save the fail.

FS is mostly linear and will give you the option of skipping a mission if you fail it enough times. Doing so will not impact your ability to finish the game... I know in FS2 it'll slightly change the later missions, but don't remember enough about FS1

The mission is horribly designed. First, it orders you to do something that is impossible (docking). Second, the scripts often get stuck and you're forced to circle the destroyer again and again, until you get noticed, whilst your drives are "malfunctioning".
Thanks to the person who suggested lowering the difficulty. That helped, so I finally passed it.

This is an easiest mission at all.
1. Put all energy on Engine.
2. Press "U" (it will lock on cargo 4).
3. Fly to cargo 4 and scan it.
4. Scan 3 newly jumped ship.
5. Scan last 3 cargos (press "U").
6. Scan all ships when they will come.
7. Scan Lucy when it will showed up. They go to Lucy's "Bay 1" (press "S" button). Ranma will fly away from it, but you need to fly inside it. Seat there and wait till Subsystem Jump will be switched on. After just press "ALT+J" and enjoy with the "no recommendation for u" end.
P.s. In the "Ener the Dragon" (previous mission) kill all shivans (except "Dragon") and Arjuna-4 team will not be in "... Judas". So it will be easy, cos Arjuna-1-2-3 always 800m-2.6km far away.
Have fun :-)

I ended up doing this on the final mission because the stupid bombers kept getting shot down.

I usually go straight for the containers. I take care of the first transports and containers. Wait for the two freighters to come to you and do the same for the 2nd set of freighters and the two big ships. The problem occurs when the demon class cruiser appears. It seems to be so long that it crosses into the path of the Arjunas. I tend to just retreat to a lower plane in order to avoid the Arjunas. The Arjunas pass right by the lucifer so their isn't much can you do except wait for them to pass if they get close to you. The lucifer just parks there waiting for you to try and enter the docking bay. I touch fighter bay 2, turn around, and high tail it out of there. Make liberal use of your countermeasures and take an erratic spiral flight path away from the huge ship. Don't panic and hit the jump drive until you are told that it is working again. Trying to activate a jump causes your velocity to drop to 0! I play the entire mission with all my power devoted to the engines. Good luck this mission is very frustrating.

Argh, can't get this mission properly completed. Any advice on scanning Eva? It just pops in and out way too fast, considering that one have to target a specific point to scan and she's DAMN FAST :>