What can we deduce about variance when we are given the noise spectral density?
Given a low-pass signal $x$ that is a realization of random variable $X$ with constant spectral density $a=N_0/2$ in $f=-B\ldots B$ (and zero otherwise) we can calculate the mean power of that signal by $\bar P = 2aB=N_0B$, where $B$ is the signal bandwidth. The mean power of a random variable is defined as the expectation of $X^2$, i.e. $\bar P = \mathrm E[X^2]$. The correspondance between mean power and variance $\sigma_X^2$ of a random variable $X$ is
$$
\sigma_X^2=\mathrm E[X^2]-m_X^2
$$
where $m_X$ is the mean value of $X$. That is, for zero mean (which is often true for noise) we get
$$
\sigma_X^2=\mathrm E[X^2] = N_0B
$$

Concerning your second question, I assume that $T_0$ is the symbol duration. Thus the symbol rate $R_\mathrm{S} = 1/T_0$. The bandwidth of a signal with symbol rate $R_\mathrm{S}$ is often approximated by $B=R_\mathrm{S}/2$ and thus
$$
\sigma_X^2 = N_0B = N_0R_\mathrm{S}/2 = N_0/(2T_0)
$$
(You can also check the units in your equation: $N_0$ is power per Hz and $T_0$ is in sec. But the result should be a power)