How are alkalide ions synthesized?
The reaction energetics

Using the method of the superposition of configurations, the electron affinity of $\ce {Na}$ was theoretically determined to be $\ce {+0.54 eV}$ $\ce {^1}$, that is, around $\ce {-52.1 kJ/mol}$. The gas-phase process

$\ce {2Na (g) -&gt; Na^+ (g) + Na^- (g)}$

has been determined to be endothermic by $\ce {4.54 eV}$ $\ce {^2}$ while the solid-state process 

$\ce {2Na (s) -&gt; Na^+ .Na^- (s)}$

has also been estimated to be endothermic by $\ce {0.8 eV}$ $\ce {^2}$.Thus, your assertion regarding the large exothermicity of the reaction is questionable. However, it is worthy to note that the $\ce { \Delta H_f}$ and $\ce {\Delta G_f}$ for $\ce {Na^+.Cry Na^-}$, where $\ce {Cry}$ = [2.2.2] cryptand, are $\ce {-10 kJ/mol}$ and $\ce {+28 kJ/mol}$ respectively $\ce {^3}$. Your large exothermic enthalpy may likely refer to the lattice energy, i.e. for the process

$\ce {M^+.Cry (g) + M^- (g) -&gt; M^+.Cry M^- (s)}$. 

For $\ce {M = Na}$, the $\ce {\Delta H}$ and $\ce {\Delta G}$ for the above process are $\ce {-323 kJ/mol}$ and $\ce {-258 kJ/mol}$ respectively $\ce {^3}$. 

Preparation of the alkalide

$\ce {Na^-}$, $\ce {K^-}$, $\ce {Rb^-}$, and $\ce {Cs^-}$ anions are stable both in suitable solvents and in crystalline solids$\ce {^3}$. The latter can be prepared either by cooling a saturated solution $\ce {^4}$ or by rapid solvent evaporation. 

The principal difficulty in preparation of crystalline salts containing alkalide ions by the method of cooling a saturated solution is the low solubility of these alkali metals in the amine and ether solutions $\ce {^3}$. Without a sufficiently large concentration of the metal dissolved in solution, precipitation of the solid upon cooling would be insignificant. This difficulty was resolved by the use of crown-ether and cryptand complexes, such as those of [18]crown-6 and [2.2.2] cryptand] $\ce {^3}$. The complexating agent complexes with $\ce {M^+}$ , shifting the equilibrium (1) far to the right, significantly increasing the concentrations of the dissolved metal ions. 

(1) $\ce { 2M (s) -&gt; M^+ (sol) + M^- (sol)}$

(2) $\ce { M^+ (sol) + Cry (sol) -&gt; M^+.Cry}$ 

This technique of using complexating agents was also what Dye et al. used in their synthesis in 1973 $\ce {^4}$. As reported by Dye et al., a sufficiently concentrated solution of sodium metal (in excess) dissolved in ethylamine with [2.2.2] cryptand was first prepared. The solution is then cooled to dry ice temperatures, giving a gold-coloured crystalline solid precipitate. Through thorough analysis, this precipitate was then determined to be $\ce {Na^+.Cry Na^- (s)}$ with $\ce {Cry}$ being the [2.2.2] cryptand.

References


Weiss, A. W. Theoretical Electron Affinities for Some of the Alkali and
Alkaline-Earth Elements. Phys. Rev., 1968, 166 (1), 70-74
Tehan, F. J.; Barnett, B. L.; Dye, J. L. Alkali anions. Preparation and Crystal Structure of a Compound which contains the Cryptated Sodium Cation and the Sodium Anion. J. Am. Chem. Soc., 1974, 96 (23), 7203–7208
Dye, J. L. Compounds of Alkali Metal Anions. Angew. Chem., 1979, 18 (8), 587-598
Dye, J. L.; Ceraso, J. M.; Lok, M. T.; Barnett, B. L.; Tehan, F. J. A Crystalline Salt of the Sodium Anion (Na-). J. Am. Chem. Soc., 1974, 96 (2), 608-609