Yes, you can use an active filter, like, as you suggest, instead of the passive first stage in the upper diagram, and instead of the nonlinear BJT-based filter in the lower diagram.
Whether the LM386 would be my amplifier of choice here: probably not, but whatever fulfills your application's needs!
You'd have to take care to:
- ensure high input impedance: the datasheet of the IC suggests (it's not a great datasheet by modern standards) that you shouldn't expect the thing to source more than 50 µA.
- AC-couple or use a rail-to-rail opamp or use a dual supply to allow the amplifier to even work with longer-term low output volume (this is the reason LM386 wouldn't be my choice)
The problem with the LM386 is that it's not an opamp; hence, calculating how you build a bandwidth-limiting loop is going to be rather intense.
If I wanted to use an LM386: a single stage RC filter would more than do the trick of being a low pass filter with a cutoff of let's say 4.5 kHz and a stopband starting at 10 kHz with 50 dB of attenuation. Either the application note/datasheet of the speech synthesizer isn't even remotely honest when it says "all noise is shifted above 10 kHz", or the filtering circuitry is total overkill. I'll go with both assumptions and say: get an opamp with 2 channels (or more). Use one channel as a voltage buffer to improve the horrendous current sourcing ability of that PWM output (whyever it is that bad – this should be a CMOS device, and a CMOS output driver should work better and could have been implemented in the same technology); use the output of that voltage buffer to feed a simple RC filter, followed by the second stage, followed by yet another RC filter, followed by the LM386. It's a stupid design (with two opamp stages, one could have built a much nicer filter), but it works without any ado.
Let's be honest here: not the route I'd take. The LM386 is a relatively crappy amplifier, and its filter characteristics are sloppy, it's hard to design around and furthermore:
You have a PWM signal. Why use a class-AB amplifier to make it loud, especially when you need to filter it at the same time?
Instead I'll propose the following:
Build a class-D amplifier. You already have a PWM signal!
Use that to drive one of the thousands of half-bridge gate drivers you can buy, for example the ADP3110A.
Use the outputs of that to control a n-channel dual mosfet device, e.g. NTGD3148N.
Connect the mosfets up like shown in the gate driver's datasheet, and AC-couple the output to an LC lowpass filter. Tadah! Class D amplifier.
Honestly, I was surprised when I read the SPO0256 datasheet! This was an impressive feat for DSP in the early 1980's.
This is essentially a decoder for LPC-compressed speech. Nice! Though I must admit: If all you want is actually to play back compressed speech, then choose any modern speech codec and do the decompression in software on the Arduino itself. All but the earliest Arduinos are plenty fast enough for simplistic codecs.
If your aim is to reproduce the original sound of the 1980's speech synths, go for it :) this is a cool project.