Yes you are right. You have degrees of freedom for the distribution. Here we have a 2-sided test, so we should be looking at a right (or left) tailed area of with degrees of freedom. Using the table below, we can extract the critical value for the desired confidence interval. (At the rows, you should be looking at and at the columns you should be reading ).

Confidence intereval for population mean. A 98% confidence interval for $\mu$ is of the form $\bar X \pm t^* s/\sqrt{n},$ where $t^*$ cuts off 1% from the upper tail of Student's t distribution with $df = n-1.$ So you are almost correct for that part. Here $t^* = 2.365.$ I get the CI $(39.00, 49.44)$ from the following brief R session.

 a = 44.22;  s = 22.0773;  n = 100
 t.c = qt(.99, n-1);  pm = c(-1,1)
 a + pm*t.c*s/sqrt(n)
 ## 38.99959 49.44041

Confidence interval for population variance. To get a 98% CI for $\sigma^2,$ one can use the fact that $(n-1)s^2/\sigma^2 \sim Chisq(df = n - 1).$ So if $L$ and $U$ cut 1% from the lower and upper tails of $Chisq(99),$ we have $$P\left(L \le \frac{(n-1)s^2}{\sigma^2} \le U \right) = .98$$ from which we get $$\left((n-1)s^2/U,\; (n-1)s^2/L\right)$$ as a 98% CI for $\sigma^2.$ In particular, for your data the CI is $(358.4, 697.0).$ (You could get $L$ and $U$ from a suitable printed table of the chi-squared distribution.) Notice that $s^2 = 487.4$ is contained in this interval, but is $not$ at the very center of the interval---because the chi-squared distribution is not symmetrical. Take the square root of both endpoints if you want a CI for $\sigma.$

 (n-1)*s^2/qchisq(c(.99,.01),n-1)
 ## 358.3833 697.0011

Note: The so-called 'rule of 30' may be roughly OK for 95% CIs because when $n \ge 30$ we have $df \ge 29$ and the values that cut of .025 from the upper tail of the t distribution are 'near' 2.0, just as 1.96 is 'near' 2.0:

 qt(.975, 29:35)  # df from 29 through 35
 ## 2.045230 2.042272 2.039513 2.036933 2.034515 2.032245 2.030108

However, this sort of approximation does not generally work well for CIs at levels other than 95%. None of the values below is really 'near' 2.326:

 qt(.99, 29:35)
 ## 2.462021 2.457262 2.452824 2.448678 2.444794 2.441150 2.437723

Please check your t table, if R code mystifies you.