I think I would start testing by having a cold clock and turning it on. I would then probe (and note whats happening) the 7410 by the regulator bottom left and the 7490 (tens of hours counter) top right in both setting and run modes. I would then wait for the fault to appear and recheck to see where the count signal is disappearing or counter not counting- to me it is a toss up between the 7490 or 7410 with my money being on the "hot" 7410.
House marked transistors are a pain- I suspect it is an ordinary Silicon transistor (not a darlington) with a gain of about 60-100, most likely NPN and capable of switching 500+mA so I would still go with something like a 2N3704.