Since half of her age is an integer, the second digit of Alberta's age must be even. When the digits are swapped, this is less than half of her age, so the second digit must be one of $0$, $2$ or $4$.
Adding $1$ to the first digit and doubling must produce the same final digit as the second digit, since this determines the final digit when the digits are swapped, 1 is added and then the result doubled. This leads to the following combinations:
||$4$ or $9$
||$0$ or $5$
||$1$ or $6$
The only combination of these that works is $52$, so Alberta is $52$.
This problem is taken from the UKMT Mathematical Challenges.