amu and g/mol relation
You are correct, but to make it a little more clear you can include the assumed "atom" in the denominator of amu:

$$
\begin{align}
m_{\ce{C}^{12}} &amp;= \pu{12amu atom^-1} \\ \\
m_{\ce{C}^{12}} &amp;= \pu{12g mol^-1} \\ \\
\pu{12amu atom^-1} &amp;= \pu{12g mol^-1} \\ \\
\pu{1amu atom^-1} &amp;= \pu{1g mol^-1}
\end{align}
$$

In other words, the ratio of amu/atom is the same as the ratio of g/mol. The definitions of amu and moles were intentionally chosen to make that happen (I'm surprised your teacher didn't explain this, actually). This allows us to easily relate masses at the atomic scale to masses at the macroscopic scale.

To check this, look at the mass of an amu when converted to grams:

$\pu{1amu}= \pu{1.6605E-24 g}$

Now divide one gram by one mole:

$\pu{1g mol^-1}= \frac{\pu{1 g}}{\pu{6.022E23 atom}} = \pu{1.6605E-24 g atom^-1}$

It's the same number! Therefore:

$\pu{1g mol^-1}= \pu{ 1 amu atom^-1}$