Putting gravel at the bottom would mainly ensure that any water pooling at the bottom would stay there longer. That is not going to be a viable solution in my opinion.
It is in theory possible to manage the situation without drainage holes but it would take a pretty high skill level (and lots of experience) to know how much and how often to water. Ideally the soil should go properly wet (to saturation or close) without any excess water beyond that point (ie. no pooling) and then not receive any water until the soil goes dry or close to dry at depth. But it would be very easy to err on the side of excess water (leading to rot) or insufficient water (leading to a slow and grueling decline by dehydration). I would not feel comfortable doing this and I have kept succulents here for 20 years. If you're relatively new to growing succulents, the probability of failure would be pretty high.
You also would not be able to leave the container out in the rain without drainage holes. And you would not be able to flush the salt from the soil, leading to a gradual accumulation of dissolved minerals from groundwater and/or added fertilizer.
So basically your best options, as I see things, would be to drill drainage holes or find a new pot.
