Are my CPU temperatures too high?
90° C is too hot. Especially since you play an old game on an i7 board. I'm guessing you are using the stock cooler which came with the CPU and you did not overclock it.
How high is ambient temperature? Do you have case fans?
Before buying a new cooler I suggest taking off the cooler, remove the old thermal paste from cooler and CPU 90 % alcohol and apply new paste. Also make sure that it is mounted correctly. On the stock cooler all push pins have to be correctly pushed through. If one is not through the board it doesn't sit right on the CPU.
Edit: I took a look around after you told me that your ambient temperature is at about 31°C and I found this: How-Ambient-Temperatures-Affect-Your-PC. In the test a better cooler was used and I think a different i7 too (you didn't tell the exact model). So reaching 90^C may not bee too far off.
My suggestions:
Change thermal paste and make sure CPU cooler is mounted correctly. This consumes the least time and money.
Add one more fan to inhale cool air or exhale heat from your case if the case has mounting points for one more.
If temperatures are still high (above 80°C) I would consider to buy a better cooler.
The temperatures do seem too high.
I agree with the other comments and answers about fans and thermal paste. However, I did not see anyone talk about the cleanliness of the system. If there is an accumulation of dust and dirt especially on the fans, the CPU cooler, or the vents, and perhaps other components, then temperatures will be higher than otherwise. The case should be opened and cleaned. If there are filters on the vents, they should be removed, cleaned, and replaced.