This is a subject that is near and dear to my heart. First of all, let me say right off the bat, the only way you are ever going to really get an upright double bass sound is to play an upright double bass. This is a topic that periodically comes up on Talk Bass. I've been a member here for over 14 years.
This is my primary instrument, a 2000 Ernie Ball Music Man StingRay unlined fretless with 3-band EQ and piezo bridge. I first bought a StingRay precisely because a StingRay with flats is the closest thing to an upright bass tone I've ever heard out of a conventional electric bass guitar, and when I made that discovery, it was with magnetic pickups only. Mine with piezo is even better. Meet 玉子 (Tamako), so named because it means "little jewel" or "jewel child" in Japanese, and also "egg" (though that is more commonly written as 卵 (tamago) ); she is Transparent Gold, the colour of a egg yolk; and my name, "Gemma", means "gem" or "jewel" in Italian:
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The first thing to understand is that there are as many "upright" tones as these are "electric" tones, so you have to consider what type of upright tone you are looking for.
I play Rotosound RS77LD strings, they are fantastic, my favourites, no need for you to change. However, should you desire to change, I would also recommend D'Addario Chromes flats or Thomastik-Infeld Jazz Bass flats.
In addition, do not overlook the D'Addario Orchestral strings like their NS Bass strings that fit electric bass guitars, or their Prelude 1/8 scale strings, which are designed for a playing length of 31 3/4", but because of the way a double bass is constructed, these should be long enough for a 34" scale bass guitar. They may actually be a little *too* long, I'm not sure, but I use 27" scale cello strings on my 30" scale Warwick. The ball end silks extend above the bridge, but this only helps improve the tone. The orchestral strings all have stranded cores, unlike the electric strings which have solid cores.
I have never actually tried nylon tapewound strings, because they are not as versatile. I may try them when I decide my current set of Rotosounds are worn out. In several years or more.
String mutes. These are a very big help, but not essential. I personally like them. If you put them all the way up against the bridge, they do not kill your sustain, and they add to the woody, thumpy character.
Piezo pickups. For this type of sound, piezo pickups are far better than magnetic. There's really no comparison. That being said...flatwound strings sound *much* better on a Precision Bass or StingRay than they do on a Jazz Bass, because of the pickup placement. Conversely, to my ears, a Jazz Bass sounds a hell of a lot better with roundwound strings than a Precision or StingRay. You can get pleasing tones from magnetic pickups. Also, if I am not mistaken, Niels-Henning Ørsted Pedersen used magnetic pickups on his upright, but that's an upright.
An "acoustic pedal". The Boss AC-2 and AC-3 Acoustic Simulator pedals were designed to help an electric guitar sound sorta kinda like an acoustic/electric, and they are terrible at that, but they are great for making piezo pickups sound better, and they do a really great job helping a piezo fretless bass with flats and mutes cop an uprighty tone. I am currently experimenting with a tc electronic BodyRez to see if that works better, but my initial impression is that it does not. It's still a nice tone, but not what I can get with my AC-2. The AC-2, however, is fairly noisy/hissy. Next I'm going to try the AC-3 and new AD-2 Acoustic Preamp pedal.
Don't listen to people who tell you to turn down the treble. Upright basses have a substantial amount of treble, and treble is essential to getting that woody/airy tone and the transients of strings knocking on ebony and finger noises. Look instead to cut upper mids, with a parametric EQ.
Play like an upright player.
Use the side of your fingers, rather than the tips. Hell, if you can, put an endpin on your bass or hold it vertically and play it like an upright. You'll be surprised how it changes your mindset. Upright tone doesn't just come from the physical construction of the instrument, it comes from the way your physical technique necessarily changes due to the vertical orientation of the instrument.
Your amplifier. Use an amp that is more like what an amplified upright player would use. I use a Mesa/Boogie Walkabout Scout 12 combo, but I would also recommend one of the Phil Jones Bass amps, which I tested pretty extensively before settling on my Walkabout. You might also investigate the Acoustic Image amplifiers, which are popular with upright players.
Well, I think I've about exhausted the advice I can give for now. Good luck, and enjoy the adventure!