Dissolve the sugar in the espresso, add the rum, and chill. Dip the ladyfingers, one at a time, into the espresso and arrange them snugly in the bottom of an 8-by-8-inch baking pan, trimming if necessary. Scoop the ice cream into the bowl of an electric mixer and beat with a paddle attachment until it's soft but still holds its shape, about 2 minutes. Spread over the ladyfingers to form a smooth surface. Freeze until firm, about 2 hours or overnight (if overnight, wrap in plastic to prevent freezer burn). Let soften slightly before scooping into bowls to serve.