The resolution of Popeye is 512x448 (horizontal screen). The sprites are done with 1:1 pixels (they are detailed), the rest is "doubled" (larger pixels). Back then, 15 KHz monitors only in arcades, so the 448 lines must be displayed in interlaced mode. MAME doesn't provide much details for the video parameters, so you have the generic "60.000000 Hz" refresh rate. In this case, the modeline is simple :
Modeline "512x448" 10.08 512 535 583 640 448 470 476 525 -hsync -vsync Interlace
525 lines, 60 Hz, 15.75 KHz.
If you have a tri-sync monitor, you can avoid interlaced display. Just double the pixel clock :
Modeline "512x448" 20.16 512 535 583 640 448 470 476 525 +hsync +vsync
525 lines, 60 Hz, 31.5 KHz.
On the other hand, Up 'n Down isn't in interlaced mode. The resolution is 512x224, so 224 lines in progressive mode. MAME lists 224x512 because the game is designed to be played with rotated monitor (vertical screen). You need to go in interlaced mode only if you want to play the game without rotating your monitor, and if it is a 15 KHz one.
What does MAME says about the timings ? Go to
system1.c :
216 #define MASTER_CLOCK XTAL_20MHz
2177 MCFG_SCREEN_RAW_PARAMS(MASTER_CLOCK/2, 640, 0, 512, 260, 0, 224)
So the pixel clock is 10 MHz, 640 pixels per line, and 512 for the active display. Then you have 260 total lines, 224 for the display.
The game shows you 256x224 pixels, but internally (for scrolling, collision detections etc.) the game use 512 instead of 256.
The timings at 15 KHz :
Modeline "512x224" 10.00 512 535 583 640 224 234 237 260 -hsync -vsync
And you get the perfect 60,096154 Hz for the refresh rate as indicated by MAME. Or, more accurately :
60,096153846153846153846153846154 Hz.

Simply put :
16640 ms per frame. That's the true number that really counts.
If you don't want to rotate your monitor (
because it's heavy and you are lazy 
), display the game in a 512 lines frame, but without going in interlaced mode, because you have a tri-sync monitor.
You should try this :
Modeline "224x512" 15.75 224 313 361 480 512 516 519 546 +hsync +vsync
Here you have the resolution of the game with the
exact same frame duration as in regular 15 KHz mode:
(480x546) / 15.75 = 16640 ms.
Horizontal frequency is 32.8125 KHz, your monitor should support it.
The complete frame is 546 lines, because if you display the 512 lines of the game within 525 lines (standard for VGA), it may be too short for the sync period. And we must extend the height of the original frame (260 lines) by a little coeficient that works well with the horizontal frequency, so at the end we keep the ratio that gave us the frame duration, and we work with an easy dotclock. Here I choose 1.05 :
520 * 1.05 = 546.
Same for the horizontal freq : the original rate of the game is 15625 Hz, so we first double it (31250) and then multiply it by 1.05.
The width of the frame is 480 pixels, so you can have a picture with a near 3/4 ratio, with black border (normally, it should be centered, but you'll need a few adjustements).
So, avoid interlaced display every time it's possible.
