Welcome to BioSE! As @Chris mentioned, transported atoms and molecules don't typically retain a record of their past history, so just by looking at a molecule in the cell you can't in general tell how it arrived in the cell (an exception might be proteins that have some chemical modification on them that allows them to be imported via a specific mechanism, such as the mannose-6-phosphate enzyme re-uptake system). In order to infer if a molecule typically enters a cell by a particular method, you will have to do several different types of experiments. I'll list a few off of the top of my head. Note to other SE contributors: since this is not my area of expertise, I'm not sure how feasible some of these experiments are, or how they are typically done, so please edit my answer to be more accurate!
- Diffusion If you want to know if something can diffuse across a membrane, then the easiest way would be to try it in vitro. One way I can imagine doing this is to use a phospholipid bilayer to separate two compartments: one with your molecule of interest, one without. By measuring the concentration of your molecule that transfers from one compartment to the other, you can see if your molecule can diffuse across a lipid bilayer. Note that this will only tell you if diffusion is possible for your molecule. It may be that cells have ways to actively prevent your molecule from diffusing across your membranes, which you wouldn't be able to know from this assay.
- Endocytosis or pinocytosis One way to assess if your molecule is taken up by endocytosis or pinocytosis would be to knock out (genetically or with an inhibitory drug) these processes in your cell and see if your molecule of interest still ends up in the cell. Depending on the context of the molecule you're interested in, you might be able to simply place cells in solution with your molecule of interest and measure the concentration of the molecule outside the cells over time. You could compare how much of your molecule is lost from solution when cells can't endocytose and compare to when they can; if your molecule stays around when cells endocytose, you have some indication that process is necessary. Note that there would have to be a lot of other controls for this experiment as well: e.g. ensuring the effect you observe is specifically caused by endocytosis being inhibited. Since endocytosis and pinocytosis form vesicle intermediates containing your particle of interest, one check to see if endocytosis might be reasonable is to see if you can see vesicles containing your molecule of interest in the cell. Of course, this requires a method for being able to visualize your molecule of interest (maybe it's fluorescent, or can be modified with a fluorescent marker? Or maybe there is a fluorescent protein that binds your molecule and can change color on binding? These things all exist, they just depend on your particular molecule and context).
Transport channels: In a similar vein, how does the uptake of your molecule change when you inhibit or remove from the cell a particular type of channel? You could measure this in a similar way as the bullet point above.
Active vs passive transport: If you wanted to see if uptake of your molecule requires active transport, you could try inhibiting ATP synthesis with a drug and see if transport of your molecule decreases over time. Note that this is in general not good for the cell (really!), so you might also have to have some more controls for this, maybe showing that transport of a molecule known to travel by passive transport is not affected on the timescale you're looking at, for example.
This list is by no means exhaustive, but I hope it gives you a sense of the type of experiments and ways of thinking you might be able to use to figure this out. Some general thoughts:
This tends to be case-by-case. While the type of experiments you might do to determine the mode of transport for different molecules are similar, the controls you will have to do to shore up your conclusions will really depend on the system involved, what type of effect a molecule has on your cells, whether your cell imports and exports this molecule, under what circumstances you might expect the cell to take up this molecule (so you have a way to trigger the cells to take up this molecule for your experiment). I don't know of a completely generic method for resolving this type of question.
Being able to measure abundance of your molecule is important. All of the experiments above rely on being able to measure the concentration of your molecule, either outside of the cell, or inside of the cell. This is often easiest if you can modify your molecule to be visualizable in a microscope or spectrophotometer, but however you do it, you need a good, reliable, robust assay in order to make strong conclusions with this assay.
Controls are really important. Thinking up all the different alternative possibilities and trying to come up with ways to rule them out is the heart of experimental science. Fortunately, it also happens to often be rewarding and thought-provoking! I certainly had fun thinking about this.
Best of luck!