DR JulieC40 - For deli-type carry-out, my husband and I like the Cranberry Deli on 45th, just west of 6th Ave. They have a space upstairs where you can "eat in," too, and we've chilled there many times.
Oh, I like Cranberry too. It can get crazy during the lunch rush, but since it's so busy, the steam table selections tend to be quite fresh.
As for being "near" Birdland, well, "near" is always a relative term in NYC. You're three long blocks away from Birdland if you're staying at the Algonquin. It's more of a 10-minute walk, rather and a 5-minute one - and, for some, that extra five minutes can make all the difference.
-Or something like that.

In any case... There are some decent dining choices in that area. There's "Little Brazil" which is 46th Street between 6th & 7th Aves. -Alas, there's really only two Brazilian/Latin places along that strip, but Margon does offer very generous lunch portions. You're also a short walk from Grand Central Station, and Grand Central's Food Court and Market. -As well as the famous Oyster Bar. The Oyster Bar isn't necessarily cheap if you're going for an entree, but a bowl of clam chowder or a clam roll will only run your about $8.00, and they even have a takeout window, although sitting at the counters is quite an NYC-experience.
You'll also be near many award-winning food carts, as well as a number of fast food and sandwich shops.
In general, I usually recommend people to venture over to 9th Ave., and just start looking at all the various cuisines, menus and price points. There's something for everyone between 42nd and 54th, and since it's off the main drags (8th Ave and Broadway), the prices and crowds tend to be smaller too.