what chemical process would take the place of photosynthesis?
On a methane world, plants would produce long chain alkanes.
The plant equivalent in this scenario would be an organism that captured ambient energy and converted it into stored chemical energy. In an oxidizing environment (like ours), oxidizing things is the easy way to release energy. Reducing things would be a way to store energy, to be released with later oxidation. On earth, plants start with oxidized carbon (CO2), capture solar radiation and convert the energy into reduced carbon (sugars).
The plants then oxidize the reduced carbon later to release the energy as they need it. Freeloader animals eat the plants and steal the reduced carbon, then themselves oxidize it back to CO2 using ample atmospheric O2.
Methane would not exist in any quantity in an oxidizing environment I do not think. On earth it gets oxidized by free O2 fairly quick. So it is safe to deduce that an environment with loads of methane would be a reducing environment. The situation there is then reversed from ours: storing energy would be via oxidation, and the release later would be via reduction.
Oxidation sounds like oxygen is involved but not necessarily.
I propose that "plants" in a methane rich environment would store nonchemical energy (you pick - solar, radiant etc) by oxidizing methane to ethane. There may be advantages to continuing this process: oxidizing further by adding additional methanes and growing a long chain alkane. This would release hydrogen (the reducer doppleganger of oxygen) and produce long chain alkanes.
The release of this energy would be via reduction: hydrogenating the alkane with H2 and releasing its constituent carbons back to methane.
Maybe methane metabolism like this occurs on earth. I have long been intrigued by the prospect that microbial action on primordial methane might be responsible for terrestrial petrochemicals - long chain alkanes. This was Thomas Gold's proposition in Deep Hot Earth. The deep earth is a methane world. There are not fossils in crude oil.
"how would they look". I have described only their metabolism. The macroscopic structure of plants has to do with energy capture (leaves), reproduction (flowers / seeds / spores) and defense against herbivory (thorns, toxins). Your methane world plants will be evolved to maximize energy capture (whatever energy they are capturing), defend against herbivores (whatever those may be) and reproduce themselves.