Pokemon transferred to Gen 2 can become Shiny because in Gen 2, shininess is determined by IVs (or "DVs"). If you transfer a Pokemon from Gen 1 that has one of eight "lucky" IV combinations, then it will appear Shiny in Gen 2. For details on how Gen 2 IVs relate to shininess, read How do Shiny Pokemon odds work?
In light of these mechanics, you could say theoretical Shiny odds in Gen 1 are the same as Gen 2: 1/8192. However, since Shinys don't exist in Gen 1, you will never get immediate feedback that you've found a Pokemon with a lucky IV combination. Further, due to limitations in how IVs are generated in Gen 1, some or all "lucky" IV combinations are outright impossible, depending on what you're hunting and where. Most notably, all the Shiny combinations are impossible for regular wild encounters in Gen 1 (sources: 1, 2, 3, 4).
By comparison, Gen 2 has no such issues, as well as helpful Shiny hunting tech such as the Shiny Ditto trick and 1/64 Shiny odds when breeding Shiny Pokemon under certain circumstances.
If you really want to Shiny hunt in Gen 1 for kicks -- assuming you're hunting a static encounter, gift Pokemon, etc. -- then you need some way to figure out if you've found a Shiny in real time. The only way to do that is to see the IVs of your encounters. There are two reasonable ways to do this:
- Check the IVs in the RAM viewer on an emulator. This would give real-time results for Pokemon of any level. Reset when the results are not Shiny.
- Use an IV calculator (option 1, option 2) for Gen 1/2 to find what stats are in the "Shiny range" for the Pokemon you're hunting. You can reset when the Pokemon's stats don't match. You will need to get the Pokemon to level 50 to remove all imprecision in the results (though the "option 2" calculator will narrow down the results earlier), so clone some Rare Candies before starting.
There is no way to improve theoretical Shiny odds in Gen 1, and no better technique than resetting the game constantly. RNG manipulation does (sort of) exist in Gen 1, but it's very unpleasant (info, info, info) and documentation for it is poor overall.