Which NN architecture solves my problem?
Welcome to SE:DataScience.

What you are looking for is called image captioning. A common approach for this is called an encoder-decoder model, where the encoder is a CNN-based NN to encode the content of an image into a vector representation, and a RNN-based decoder generates its caption conditioned on that vector.

This is a relatively well-studied area, with a lot of well-formatted dataset and open-source implementations, for example in TensorFlow. The link above can also serve as a good tutorial for this topic.

More recent models can use the attention mechanism to focus on specific part (object) in an image, do dense captioning, or answer questions about images, etc.