TL;DR: There is evidence both that 5:2 intermittent fasting diet increases activation of SIRT1, and that this gene allows for various health benefits, including fat regulation, insulin reduction (anti diabetic) and some anti-ageing properties.
N.B: I limited the answer strictly to the scope of the health benefits as found in the OP's claim (specifically fat storage, aging and DNA regulation), but there are very likely more effects to consider.
An article at ScienceAlert.com describes a study by the University of Florida and Harvard Medical School, alternatively eating 175% of their usual caloric intake during 5 days and 25% the other 2 days with a small sample of healthy participants (N=24). The article can be found here. Interestingly enough, it appears that it did increase SIRT1 expression, but that the gene activation alone isn't a sufficient factor to result in beneficial health effects:
The researchers found that the act of intermittent fasting was causing
a slight increase the SIRT3 and SIRT1 genes, which encode proteins
called SIRT3 and SIRT1. These proteins belong to a class of proteins
called sirtuins, which previous experiments have shown can extend the
lifespan of mice if their levels are increased. But here’s the catch -
they only work in this way in response to oxidative stress, and the
team found that they only increased during the three weeks were [sic]
antioxidants were avoided. When the participants started taking the
supplements, these increases disappeared.
This indicates that antioxidants may restrict the effects of SIRT1.
Ok, so there is evidence that the 5:2 intermittent fasting diet increases SIRT1 expression. What about the effects of SIRT1 on humans? From the same article, there are hints that the diet reduces insulin levels:
Interestingly, the team found that the increase in SIRT 1 and 3 genes
appeared even in the absence of weight-loss, which suggests
weight-loss isn’t the trigger here. They also found that levels of
insulin dropped as a result of the diet, which means it could also
have an anti-diabetic effect. They explain in the journal Rejuvenation
However, the sample size of the study is quite small (N=24), as such the conclusions should be taken into consideration but with caution.
Another study investigated the effects of diet on longevity, which can be found here and itself references 2 other studies indicating expected health benefits of SIRT1:
Systemic or tissue specific overexpression of some sirtuins (i.e.
SIRT1, SIRT3 and SIRT6) also increases genomic stability, reduces
NF-kB signaling, and improves metabolic homeostasis through histone
deacetylation (Mouchiroud et al., 2013; Guarente. 2013). Moroever,
activation of SIRT1 and AMPK activates PGC-1a, a transcriptional
regulator of mitochondrial function, antioxidant defenses, and fatty
acid oxidation (Wu et al., 1999).
As such, there is evidence that SIRT1 allows some form of DNA maintenance & repair and fat storage regulation.