You would need to change trains in Munich, but the journey seems very doable to me.
For time efficiency, there is a night train option which requires only about half a day of travelling during the daytime; alternatively it's possible to do the whole thing in a day with day trains only but it takes up the vast majority of a day.
Outward journey
Night train and day train
You can catch the EN462 night train from Budapest Keleti to München Hauptbahnhof leaving 20:40 on the evening of the 2nd September and arriving 06:29 on the morning of 3rd September. This train has full sleeping coaches (which can be reasonably expensive), cheaper couchette coaches which allow you to lie flat but where you're mixed in with other people and where people usually remain clothed, and seated coaches if you're really on a budget.
From there you can catch a direct daytime ICE train, ICE726, leaving München Hauptbahnhof at 07:26 and arriving Essen Hauptbahnhof at 13:13.
The price of all of this will depend on which accommodation you decide upon for the overnight leg.
Day trains only
On 2nd or 3rd September you can catch the RJ62 train from Budapest Keleti at 09:40, this gets you into München Hbf at 16:33 where you have plenty of time to change onto the ICE 512, the 17:25 service direct to Essen Hbf, arriving 22:59. Both will be nice, clean, modern trains; one-way it seems to cost about 90-125 EUR in 2nd class.
Overnight (non-sleeper) train option
Another alternative is to catch a day train from Budapest to München, then an overnight non-sleeper train to Essen. This will mean spending the night in a seat, so you might very reasonably decide this is not for you, but the option is there and it's quite cheap. Leaving at 15:40 on Friday 2nd on RJ68 will get you into München Hbf at 22:31, where you have time for a late dinner before your 00:01 overnight service ICE618 to Essen Hbf, arriving at 06:58. The whole thing is only showing as costing around 60 EUR right now.
Note there would normally be yet another option to travel on a sleeper train from München Hbf to Köln Hbf and catch a short day train from there to Essen Hbf, but it doesn't seem to be stopping there at the time you want to travel, probably due to engineering works (track work).
Return journey
A day train followed by a night train
Similar to the above, you can leave Essen at 17:00 on the ICE 613, changing at München Hbf onto the EN 463 arriving Budapest at 09:19.
Day trains only
There is a similar option involving leaving at 11:00 on the ICE 517, and having just over an hour at München Hbf to change onto the 17:30 RJ261 arriving ultimately at 00:19 the following morning. There also a much earlier option leaving at 04:27 in the morning, but obviously costing less. I can't find a similar overnight option with non-sleeper trains as I gave above for the return journey though.
Where to check ticket prices
I've been using the Deutsche Bahn journey planner for this. It will let you buy tickets for trains within Germany, and also the RJ express day trains. For the night trains, I believe you should be able to buy tickets again for the whole journey at ÖBB's website.
Travelling from Düsseldorf Airport to Essen Hbf
If you choose to fly instead, there are many direct Regional Express trains from Düsseldorf Flughafen to Essen Hauptbahnhof, running approximately every 15 minutes, and taking approximately 20-30 minutes depending on the train. They run on routes RE1, RE2, RE6, and RE11. You can travel on these with a fare of 13 EUR each way, bought from VRR (the authority in charge of transport in that region).
