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What do you do with notes below the E on a 4 string?

For example here is a professionally-engraved Hal Leonard arrangement for the classic "Earth Angel." Notice all those low Eb's and C's just in the first measure. If you didn't know bass was an octave-transposing instrument, you'd totally think you'd need a 5-string to play all those old doo wop hits of the 1950s. My first teacher totally dropped the ball on teaching me this, and it totally blew my mind when I realized it on my own, after struggling with sheet music like this for years. My friends would bring songbooks to band practice, and I'd be like, uh-oh, sorry guys, I don't have a 5-string, but I'll do my best!

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With that kind of music sheet I’ll just follow the chords and not read the music
 
I play what I have. I put hipshot d-tuners on all my four string basses so I can go down to D if I need to. Otherwise I play the note an octave up. If the deep tone is really important I’ll either use an octave pedal or play a five string.
 
If that Eb or D re-occurs a lot Il use my de-tuner. Otherwise, go up an octave.

I've done Jason Aldean's "She's Country" and "Hicktown" for about 6 years with a 4 stringer (both are in C#). On She's Country, I play the lines up an octave when necessary. We play the other in D, and I detune the E string to D.

No one has ever complained...
 
If you are sight reading or playing something with Eb or lower but you only have a 4 banger what do you do?

Play an octave up, a chord tone, or add a rest note?

Thanks!
Drop the tuning, if it isn't already dropped. Some of my basses are tuned standard, some are tuned half a step down, and one is tuned standard with an adjustable hipshot drop tuner tuned to D on the E string. Or I can grab my fiver.

In the 3 piece, all originals project that I'm involved with, we keep to standard tuning. But before Covid when I was playing out a lot with other musicians I was tuned half a step down 99.9% of the time because a lot of vocalists want it that way.
 
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Generally, octave up. Don't think I'd ever just rest. Sometimes, if taking a note an octave up, it makes sense to take a few of the notes around it an octave up (even if they're within your range) to better preserve the continuity of the line (i.e. switch octaves in a more musical way). This, like a lot of things, is context dependent.

Not sight-reading, but if I'm playing my own line, I may well drop to another chord tone (say the G 3rd of an Eb Major chord). Sometimes this is more musical/more interesting even if you have the low Eb on tap. Double-stops/chords can work well, as someone mentioned. Again, all context-dependent.

Finally, as several people have mentioned, when reading off of piano (or most other) parts, it's worth remembering that the bass sounds an octave lower than written. Still, most legit bass transcriptions should have the part notated properly for bass (this includes orchestral parts), though I seem to recall some exceptions.
 
Depends. If you can, use your ear for reference. What do they do in the recorded version? Sometimes, as Mushroo points out, it's a transcription thing and the octave is what was actually played. Sometimes it's a true lower Eb, but the octave will do. But if it's a low, droning Eb like, for example, the Eb that plays throughout the song Superstition, you probably want to downtune your bass.
 
Often, I just play the Eb an octave up.
In my cover band, I would tune my E string down to Eb for a few songs: Superstition and Sledgehammer.
It’s not a big deal as long as you’re prepared. Think of it as part of your bass player bag o’ tricks.
 
If you are sight reading or playing something with Eb or lower but you only have a 4 banger what do you do?

Play an octave up, a chord tone, or add a rest note?

Thanks!
Detune the E string to a D and carry-on...don't forget to tune back up for the next song (been there, done that multiple times, don't recommend it). If you find yourself needing to do that on very many songs, buy a 5 string so you can just play without tuning down and up again. For me, 3 songs a night was a pain and 4 songs a night made me buy a 5-string.

Another thing you can do is buy a Hipshot Drop D-Tuner, which will allow you to flip a lever up at your E tuning peg to switch the string back and forth between the E and the D. Then if you forget to switch it back to E, you can do it very quickly and accurately but just flipping that lever at your first opportunity.
 
Octave up is a one-size-fits-most option available to all of us 4-stringers.

For song-to-song changes, this works:
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Digitech Drop. But that is potential disaster if anyone forgets to turn it off for songs in standard tuning, so we don't change tunings during sets.

I use it mostly when practicing to recordings of songs that we are covering. It minimally affects tone if going up or down a step or so, but increasing artifact does change tone the farther you go from standard. I don't care for the octave tones, but it is really useful for my practice purposes.

We do everything a half-step down, so I keep my No. 1 tuned and set up for that tuning. So there's a 3rd option.

A drop D tuner is very likely to show up in my stocking this year. I've been a good boy this year, so I really have my hopes up! That's option 4.

And the fifth option, which I hope will someday be forthcoming: a fiver, which will present another option someday. Gassing for one, but funds are tight. And although I was a good boy this year, I am grading on the curve a little. Pandemic and all, you know.
 
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I don't run into this often but when I do I just play the lowest note in a different place on the neck.
I have run into de-tuning guitarists. In which case I have de-tuned or brought two basses, one standard and one de-tuned. If it's only a few songs I just play the song with a standard tuned bass. It's not that hard! I have also tuned my bass B E A D but never really liked it.