It isn't whether or not the meter is digital or analog, but whether it has high enough input impedance to avoid loading the circuit and causing a false reading.* A really high quality analog meter will probably work, as will a FET (transistorized)VOM, or a VTVM (vacuum tube voltmeter), but a cheap multimeter may not. If yours isn't a pretty good one, you might be better off getting a cheap digital meter, which can be had these days for under $20, and sometimes under 10.
* remember that an analog meter movement is just a little motor, constrained by a spring from rotating very far. The circuit you're testing powers that little motor, pusing it until it stalls against the spring tension. Some solid state circuits just can't put out the current without voltage drop, and the TPS circuit is probably one that can't.
If the TPS is good, it should stay adjusted for a long time, but if it's bad, adjusting won't do much good. In my experience, the bolts are more likely to seize than to come loose. Since you have an analog meter, one thing it's very good at is detecting glitches in a variable resistor - something you can't readily do with a digital meter. With the TPS unplugged, probe its connectors with the ohmmeter until you find a pair of terminals that give variable resistance when you open and close the throttle. Do this slowly, watching the ohmmeter, and make sure that the needle moves evenly, and doesn't jump or drop at any point. That's a symptom of a bad TPS.