Lack of shaker is fine, in fact Martini's should be stirred, James Bond was just trying to be different. But you do need a way to strain the ice from the drink. I use a shaker, but just swirl it to mix rather than shaking, because it has a convenient strainer top. But pro bartenders generaly stir with a long spoon in a pint glass, and then use one of these strainers on top of the glass for pouring:
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Martini glasses can be Amazon'd. They even have reasonably nice looking plastic ones now, if your drinking tends to be poolside and you want to avoid broken glass around the pool due to kids throwing balls around. DAMHIKIT
The rule on Martini glasses is bigger is better, those little old 2 oz. jobbies grandma bought in 1940 don't really work. You want a 4 oz. or larger glass.
Tequilla Martini (it's simple, but it's big!):
2.5 oz. of your favorite top-shelf Tequilla
1.5 oz. Carpano Dry Vermouth (no point ruining good tequilla with cheap vermouth)
2 large crisp Mezzeta martini olives (pre-soaked in vermouth)
So, obviously a 4 oz. drink, which is about twice the size of the original Martinis served 80+ years ago. I actually like 1800 Resposado tequilla, but most will say it should be made with Blanco. Hey... whatever floats your boat.
Gotta help mom move in 95F heat this afternoon, with our usual mid-Atlantic humidity in spades. Sister is there supervising the movers who are doing the heavy lifting this morning, I'm second shift to help them set up the new place this afternoon into evening. Maybe I'll recover with a Tequilla martini, when it's all done.