Both BER and BUD are great...
I have been to both - Berlin for a weekend, and BUD for many, many weekends.
If you go to Berlin - the history is amazing. One of the best ways to see the city is on one of the walking tours - do this on the first touring day. I did the 'insidertour' - which was fantastic, but there are others (all claim to be 'the best' - but ALL have good write-ups in the guidebooks).
The 'insidertour' we did had an advertised walk time of three hours - although we didn't finish it under four. The guide takes their time and there really is so much to see.
The guides are all fluent in English and are Berliners.
If you have a choice of airport to fly into - Templehoff is the one - an amazing piece of art deco architecture - right in the town centre. Well worth a visit in its own right.
On the other hand - Budapest is amazing for its ambience and its thermal baths and amazing scenery - perfect for relaxing after a trip to Egypt. I always stay at the Gellert Hotel (via the Danubius hotels website). As a guest you simply walk downstairs for free entry. (Warning - the Gellert, built in 1912 (I think) is faded granduer! I love it - but others wanting internet access in the room and other mod cons always prefer to stay at one of the western hotels) - but check out the Gellert website).
In BUD make sure you go and have a tour of Parliament.
A single room there is Euro66 per night and is adequate. That includes breakfast. I have also stayed at the Inter.Continental in BUD - but you get a standard western room (although very nice!).
On arrival in BUD - taxis are available from the front of the airport. They are SET PRICE (5600 forints). You get a chit from the guy before you get in. The airport bus takes forever (and only leaves when full - so some freinds of mine have waited 45 minutes to leave). The cost for that is 2100 forints. Your chit when you get into the taxi at the airport contains a set price for returning to the airport (your hotel books it via the number on the back of the card) - the cost is 3000 forints.
Be warned - there are touist rates for taxis in town itself - you can pay BIG buck for a short journey. Always make sure you get into a taxi that has been booked by your hotel, or restaurant etc etc.
Any questions, please let me know,
Regards
WG