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Any chemical fertilizer, whether it says "organic" or not, carries relatively the same amount of "safety" for use. Organic fertilizers (such as the liquid form made by Miracle Grow, which is what I use) carry concentrates of several compounds that help plants develop in soil that may be lacking ideal nutrients.

That said, if your tomato plant is in a pot, there really should not be ANY runoff - if there is, you're over-watering it.

To check if your plant needs water, use a pencil, dowel-rod or similar tool to make a test bore about one and a half to two inches deep in the soil. Feel around the inside of the hole with your finger - if the soil is damp under the surface, it's good to go. If it's starting to feel dry and crumbly, it needs water. (Soil can sometimes look "dry" directly on the surface, and still have moisture underneath, especially if it's in a sunny area or under a lamp.)

You can also get moisture gauges to test the soil electronically - I've had off and on luck with them. I am much more a hands-on, direct-attention gardener.

Tomatoes like sun and water, but be careful not to overdo both! Keep an eye on the leaves daily, and watch for signs of wilting, yellowing, or dropping - these indicate a health problem with the plant, and the most common causes are under-watering a plant in a sunny location or over-watering a plant in a shady location.

You'll find tomato plants to be a completely different experience from cacti, but I think they're absolutely wonderful. The plants themselves are leafy and have a strong fragrance, particularly as they mature, and having fresh tomatoes beats ones that have traveled by truck to a supermarket any day. (Many of those also often have a red-orange tint rather than a strong red one, not because of the variety of tomato, but because they were picked early and/or artificially ripened. If you let your tomatoes ripen on the plant, you'll find that they're a robust, vibrant red (unless you have a specialty variety such as zebra, lemon, etc) and have a richer taste and texture when you eat them.)

The tomatoes that I'm growing have proven to be reeaalllyyy resilient. we've had three or four big storms that have toppled over my plants, but they always pop right back up.

Tomato plants are indeed resilient! One of mine was knocked over on accident and lost about the top third of the plant, but it's since rebounded.

One of the things you can do with tomato plants, especially cherry tomatoes, is to selectively prune some of the leaves so that the plant isn't wasting energy that could better go to producing fruit or larger branches. If you look closely at your plant, you'll see that some of the branches and secondary branches have a fork shape, where it splits into two smaller sections. Frequently, a tiny little branch, barely noticeable, with tiny little leaves, will be growing in or very close to the fork. Reach in and clip those off. (They are very soft - if you have long fingernails, your fingernails are often sharp enough; otherwise, use a very small pair of pruning shears or small, sharp scissors so that you don't accidentally cut the main branch.) By clipping off those tiny sprouts, the tomato plant will direct the energy, water and nutrients that were going to it to other parts of the plant. It doesn't seem like a lot when you clip just one or two, but if you keep after it as your plant grows it adds up. After a little while of doing it, you'll start to notice bigger leaves, heftier stems and branches, and more blossoms.

Sometimes you can get a tomato to root from clippings without using any rooting powder or rooting hormone. If another piece of the vine breaks off, try sticking the broken side in the soil. The worst that could happen is that it doesn't grow and you have to take it out later. On the other hand, you might get a second plant out of the bargain.
The biggest nutrient deficiency I have experienced with my potted tomatoes is calcium deficiency, which results in blossom end rot. You know you have this problem because large rotten-looking brown spots will form on the ripening tomatoes. By the time you see the symptoms, it is normally too late to do much about the condition for the season. It is easily preventable by mixing a couple of tablespoons of gypsum into the soil, or the shells of a few eggs into the soil. Depending on the fertilizer you use, you may not need this. I generally grow my tomatoes organically, though.

Cool! I like gardening and taking care of our many trees and plants. I too like to name my plants 3nodding Thats cool that you have a tomato plant. Good luck with your plant!