Hello there,
Whether or not it works is not a matter of hopefulness, it is a matter of drive specifications for both the driver and the driven.
Since the driven is a 7400 series device, it should have a 7400 device to drive it. What matters most here in this special case is just how many inputs are being driven. If it is just one input being driven it might work, pending the investigation of the low and high state requirements of the driven.
There are writeups on this but i don't know where they are right now. They also go by 'fanout' meaning if a drive chip has a fanout rating of 5 then it can drive 5 inputs of the same actual family. Someone could check but i think the LS series is 1/2 of the regular series, so that if the driven inputs only total 2 and the fanout of the driver is 4 but of the lower LS series, it should still work.
Now if the driven inputs are 4 and the fanout of the driver with the LS series chip is only 4, then it's not a good idea to use that LS chip even if it seems to work at first. That's because the chip specs are for a given temperature range so it may work one minute and not work the next.
It's not hard to find a 7490 chip. If you use that you can be sure it will work, no question about it, unless something else is wrong too.
As someone mentioned, the copper clad may get ripped up for some pins, which is not really too unusual. You can be careful but just in case there is a known fix. Since the entire trace does not rip up, only near to the pin holes, you can get a AWG #26 gauge bare wire and make a small loop around the pin of the new device, then run the tail down the part of the trace that still remains. You then solder both the pin again to the loop, and the tail to the copper that remained during removal of the old chip. Of course be careful you do not create any solder bridges. The AWG #26 wire is thin enough to go around a DIP pin without touching any other pins. I dont think i would try this with AWG #22 though, although AWG #24 may work too.
If you cant get #26 wire sometimes you can pull a strand out of a piece of regular stranded wire. It should be thin like AWG #26, but you can test it by wrapping it around one pin and make sure it is thin enough so that it does not touch any other pins, before and after soldering.