Relationship between the Lagrangian and consumption euler equation?
In this simple problem, using a Lagrangean is an overkill, direct substitution of the constraint is perhaps better.  

Anyway, treating the two consumption levels as two distinct decision variables under the budget constraint, the Euler equation emerges from the combination of the two first-order conditions:

$FOC$'s

$$\frac {\partial \mathcal L}{\partial C_1} = 0 \Rightarrow \frac 1{C_1} = \mu$$

$$\frac {\partial \mathcal L}{\partial C_2} = 0 \Rightarrow \frac {\beta}{C_2} = \mu\frac {1}{1+r_1}$$

Substitute the first into the second

$$\frac {\beta}{C_2} = \frac 1{C_1}\frac {1}{1+r_1}$$

re-arrange and you 're done.

I would suggest to also do that with a general utility function, where in the Euler equation the first derivatives of the utility will appear.