If you are willing to tolerate the B button just resetting it, you can do it with one chip, the 4017 decade counter. I haven't read every link in every post above and it's almost certain to have been mentioned.
It has a clock input to which you connect your "up" button, a reset input to which you connect your "Reset" button, and ten outputs, to which you connect your LEDs (always via a suitable series resistor).
There are subtleties. To turn it into a bargraph, you will need to dance the funky chicken with some transistors (or gates) and diodes to ensure that everything less than the active output turns on, because by default it just has one output active. You will need to debounce at least the "up" switch, or you'll find that switch bounce will make it count up in units of more than one. This is easily googled.
Option B involves something like a LM3914 bargraph meter driver. It's a bit more complicated to set up, but it's basically a bargraph volt meter. You'd want to make it read the voltage across a capacitor, and have your A and B buttons charge the capacitor via a resistor. The graph would increase while you held one button down, and decrease while you held the other one down. It would be nonlinear, possibly very nonlinear, so the speed with which the segments illuminated would change as it got near the ends.
Neither of these really replicates the behavior of the screen prop, which I suspect used a microcontroller (which is how I'd do it). That said, if you're not into microcontrollers, I think it can be done in discrete logic, probably two chips.
Chip one is a 4510 BCD counter. This operates broadly like the 4017, except it has a count-up input, and critically, a count-down input. The other difference, though, is that it doesn't have a completely decoded output. It outputs BCD, binary coded decimal, which is basically four-bit binary data. That's no disaster, though, because you could then use a 74HC42, which takes the four BCD outputs from the 4510 and decodes them to individual outputs.
You would still need to debounce the switches and do something to ensure that all the segments below the one you selected were lit, which could actually get quite sticky - I'd do it with transistors and diodes, probably. At that point you would have exactly replicated the behavior of the screen prop, with the possible exception that if you went all the way up to the top and kept pressing "up" it would probably wrap round to the bottom again.
You could I suppose have a 4017 connected to a high-speed oscillator, and connect a series of AND gates between its outputs and the outputs of a 74HC42, with their outputs connected to the reset on the 4017. That would keep all the segments below the selected segment lit. More could be used to gate off the "up" and "down" buttons to prevent overflowing. But that's ten AND gates, and now the thing is becoming a five or six chip monster.
Which is why people do it in microcontrollers.
HF