Hello Fryderyk, I am not sure just how high protein you are looking to achieve in your diet (for reference, I am a small woman and I am ok with about 50-60 grams of protein per day), but I do think you can achieve it with a mix of eggs, dairy (cottage cheese and greek yogurt are high protein), nuts/seeds/legumes.
As far as the cost of being vegetarian or vegan, it CAN be less expensive, but it all in how you plan it. Buying dry beans from the bulk section of a grocery store is very cheap, much cheaper than canned beans, but it requires time in preparation. If you are looking for cheap and good quality food, you will have to invest time in the kitchen. Any more convenient foods, like Quorn, will be more expensive, and may also be lower quality nutrition (somewhat of a junk food).
If you are willing, growing your own green beans on even an apartment balcony can provide cheap and healthy food source of some protein, vitamins, and minerals. You have to find a what works for you. And yes, my love Barachiel is correct about healthy fats. Include things like coconut, avocados, flax or fish oil. I like using whole food sources as opposed to pill supplements because I love the taste of those foods, and whole foods will give you a variety of minerals, vitamins, fiber, and other nutrients.
Bottom line, always a trade off between cost, quality, and convenience.